tangent ratio practice and problem solving modified

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High school geometry

Course: high school geometry   >   unit 5.

• Triangle similarity & the trigonometric ratios

Trigonometric ratios in right triangles

• (Choice A)   3 5 ‍   A 3 5 ‍  
• (Choice B)   4 5 ‍   B 4 5 ‍  
• (Choice C)   3 4 ‍   C 3 4 ‍  
• (Choice D)   4 3 ‍   D 4 3 ‍  

Tangent Ratio

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Table of Contents

Last modified on August 3rd, 2023

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Tangent ratio.

The tangent ratio is one of the trigonometric ratios for right-angled triangles. It is the ratio of the opposite side to the adjacent side concerning an angle.

Consider a right triangle ABC, where AC is the hypotenuse and AB and BC are the other two sides of a right triangle. Thus, for any angle θ in a right triangle,

${\tan \theta =\dfrac{Opposite side}{Adjacent side}}$

Precisely, ${\tan A=\dfrac{BC}{AB}}$ and ${\tan C=\dfrac{AB}{BC}}$

Thus, tangent ratios can be used to calculate the angles and sides of right-angle triangles, similar to the sine and cosine ratios. Furthermore, it is possible to find the tangent ratio given one angle of the right triangle other than the right angle.

Considering the tangent ratio ${\tan A=\dfrac{BC}{AB}}$, if the measure of angle A is known, it is possible to find the tangent ratio of angle A. Again, finding the tangent ratio is easy if one side and the hypotenuse are known. For example, if AC and BC are known, then by using the Pythagorean theorem, we get

${\left| AB\right| =\sqrt{\left| AC\right| ^{2}-\left| BC\right| ^{2}}}$

Thus, ${\tan A=\dfrac{\left| BC\right| }{\sqrt{\left| AC\right| ^{2}-\left| BC\right| ^{2}}}}$

Finding the Tangent Ratio

Consider 3 right triangles with different side lengths but the same angle measuring θ. Let us now find the tangent ratios for all three triangles.

For the smaller triangle,

tan θ = ${\dfrac{4}{3}}$ = 1.34

For the medium triangle,  

tan θ = ${\dfrac{8}{6}}$ = 1.34

For the large triangle, 

tan θ = ${\dfrac{12}{9}}$ = 1.34

Thus, the tangent ratio remains the same regardless of the size of the right triangle. This specific ratio is the trigonometric ratio of the triangle for an acute angle θ, and it applies to similar triangles.

For example, tan 45° = 1

If one angle in a right triangle is 45°, the ratio of the length of the opposite leg to its adjacent leg is 1.

The tangent ratio is thus a function that takes different values depending on the angle measure. We can measure an angle in degrees or radians. It can also be calculated using a calculator.

Solved Examples

Now that we learned the concept let us solve some problems.

tan P = ${\dfrac{QR}{QP}=\dfrac{12}{5}}$ tan R = ${\dfrac{PQ}{QR}=\dfrac{5}{12}}$

As we know, tan 37° = ${\dfrac{x}{12}}$ tan 37° ≈ 0.75 => ${\dfrac{x}{12}}$ = 0.75 x = 9

Find the angle with a tangent ratio of 0.4877

tan -1 (0.4877) = 2525.99849161 ≈26 degrees

Find the angle with a tangent ratio of 0.9325.

tan-1(0.9325) = 42.9995 ≈ 43 degrees

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Sine, Cosine and Tangent

Opposite & adjacent sides and SOHCAHTOA of angles

This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle.

The Sine , Cosine and Tangent functions express the ratios of sides of a right triangle .

To which triangle(s) below does SOHCAHTOA apply?

SOCHAHTOA Video Lesson

Interactive Angles SOHCAHTOA

Try activating either $$ \angle A $$ or $$ \angle B$$ to explore the way that the adjacent and the opposite sides change based on the angle.

The Sine Ratio

Answer: sine of an angle is always the ratio of the $$\frac{opposite side}{hypotenuse} $$ .

$ sine(angle) = \frac{ \text{opposite side}}{\text{hypotenuse}} $

$$ sin(\angle \red L) = \frac{opposite }{hypotenuse} \\ sin(\angle \red L) = \frac{9}{15} $$

$$ sin(\angle \red K) = \frac{opposite }{hypotenuse} \\ sin(\angle \red K)= \frac{12}{15} $$

Remember: When we use the words 'opposite' and 'adjacent,' we always have to have a specific angle in mind.

Range of Values of Sine

For those comfortable in "Math Speak", the domain and range of Sine is as follows.

• Domain of Sine = all real numbers
• Range of Sine = {-1 ≤ y ≤ 1}

The sine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key sine values that span the entire range of values.

The Cosine Ratio

The cosine of an angle is always the ratio of the (adjacent side/ hypotenuse).

$ cosine(angle) = \frac{ \text{adjacent side}}{\text{hypotenuse}} $

$$ cos(\angle \red L) = \frac{adjacent }{hypotenuse} \\ cos(\angle \red L) = \frac{12}{15} $$

$$ cos(\angle \red K) = \frac{adjacent }{hypotenuse} \\ cos(\angle \red K) = \frac{9}{15} $$

Range of Values of Cosine

For those comfortable in "Math Speak", the domain and range of cosine is as follows.

• Domain of Cosine = all real numbers
• Range of Cosine = {-1 ≤ y ≤ 1}

The cosine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key cosine values that span the entire range of values.

The Tangent Ratio

The tangent of an angle is always the ratio of the (opposite side/ adjacent side).

$ tangent(angle) = \frac{ \text{opposite side}}{\text{adjacent side}} $

$$ tan(\angle \red L) = \frac{opposite }{adjacent } \\ tan(\angle \red L) = \frac{9}{12} $$

$$ tan(\angle \red K) = \frac{opposite }{adjacent } \\ tan(\angle \red K) = \frac{12}{9} $$

Practice Problems

Highlighted problems.

In the triangles below, identify the hypotenuse and the sides that are opposite and adjacent to the shaded angle.

Hypotenuse = AB Opposite side = BC Adjacent side = AC

Hypotenuse = AC Opposite side = BC Adjacent side = AB

Hypotenuse = YX Opposite Side = ZX Adjacent Side = ZY

Hypotenuse = I Side opposite of A = H Side adjacent to A = J

No Highlights (harder)

Identify the hypotenuse , and the opposite and adjacent sides of $$ \angle ACB $$.

First , remember that the middle letter of the angle name ( $$ \angle A \red C B $$ ) is the location of the angle.

Second: The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle -- which in this case is the red angle in the picture.

Identify the hypotenuse , and the opposite and adjacent sides of $$ \angle RPQ $$.

First , remember that the middle letter of the angle name ( $$ \angle R \red P Q $$ ) is the location of the angle.

Identify the hypotenuse , and the opposite and adjacent sides of $$ \angle BAC $$.

First , remember that the middle letter of the angle name ( $$ \angle B \red A C $$ ) is the location of the angle.

Identify the side that is opposite of $$\angle$$IHU and the side that is adjacent to $$\angle$$IHU.

First , remember that the middle letter of the angle name ( $$ \angle I \red H U $$ ) is the location of the angle.

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Curriculum  /  Math  /  10th Grade  /  Unit 4: Right Triangles and Trigonometry  /  Lesson 12

Right Triangles and Trigonometry

Lesson 12 of 19

Criteria for Success

Tips for teachers, anchor problems.

• Problem Set

Target Task

Additional practice.

Describe the relationship between slope and the tangent ratio of the angle of elevation/depression. Use the tangent ratio of the angle of elevation or depression to solve real-world problems.

Common Core Standards

Core standards.

The core standards covered in this lesson

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

Expressions and Equations

8.EE.B.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Describe how the slope of a line compares to the tangent of the angle of depression.
• Use the context of the problem to determine whether the angle is an elevation or a depression. 
• Explain that the ratios of any right angle formed from a single line where the line is the hypotenuse will be equal (slope triangles from 8th grade). 
• Describe that the angle of elevation is congruent to the angle of depression, but they are not the same angle.

Suggestions for teachers to help them teach this lesson

• This lesson is an extension of 8.EE.6 by relating slope to the tangent ratio. As a result, students may need to review the concept of slope before they can fully access this lesson. It is recommended to spend time outside of class building this skill. 
• This is the first lesson out of two that begins to have students model real-world examples of trigonometric ratios.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Brandon and Madison use different triangles to determine the slope of the line shown below. 

Brandon started at (0,-1) and drew a right triangle going up 2 units and right 3 units.  Madison started at (-3,-3) and drew a right triangle going up 6 units and right 9 units.

• Draw and label both triangles on the graph. 
• How can Brandon and Madison use their triangles to find the slope? 
• What trigonometric ratio do Brandon and Madison use to find the slope? 

Guiding Questions

Below is a line segment on a coordinate grid.

• Draw the right triangle that you can use to find the length of this line segment. 
• Find the length of the line segment. 
• Find the measure of all of the angles of the triangle formed. 

Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20°. The lighthouse is 28 meters tall and sits on a cliff 45 meters tall as measured from sea level.  What is the horizontal distance between the lighthouse and the ship?

Geometry > Module 2 > Topic E > Lesson 29 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Samuel is at the top of a tower and will ride a trolley down a zip line to a lower tower. The total vertical drop of the zip line is 40 feet. The zip line’s angle of elevation from the lower tower is 11.5°. Sam’s friend is not going to zip-line but wants to walk along the ground from the tall tower to the lower tower. How far will Sam’s friend walk to meet him?

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include word problems where students are required to draw a diagram with the angle of elevation or depression.
• EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 29 — Example 1, Example 2, Exercise 2, and Problem Set
• Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry — Lessons 10 and 11 (Focus on the questions regarding angle of elevation and depression)
• CK-12 Angles of Elevation and Depression

Topic A: Right Triangle Properties and Side-Length Relationships

Define the parts of a right triangle and describe the properties of an altitude of a right triangle.

G.CO.A.1 G.SRT.B.4

Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.

Define the relationship between side lengths of special right triangles.

G.SRT.B.4 G.SRT.B.5

Multiply and divide radicals. Rationalize the denominator.

A.SSE.A.2 N.RN.A.2

Add and subtract radicals.

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Topic B: Right Triangle Trigonometry

Define and calculate the sine of angles in right triangles. Use similarity criteria to generalize the definition of sine to all angles of the same measure.

Define and calculate the cosine of angles in right triangles. Use similarity criteria to generalize the definition of cosine to all angles of the same measure.

Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°. 

Describe and calculate tangent in right triangles. Describe how the value of tangent changes as the angle measure approaches 0°, 45°, and 90°.

G.SRT.C.6 G.SRT.C.7

Solve for missing sides of a right triangle given the length of one side and measure of one angle.

Topic C: Applications of Right Triangle Trigonometry

Find the angle measure given two sides using inverse trigonometric functions.

Solve a modeling problem using trigonometry.

Topic D: The Unit Circle

Define angles in standard position and use them to build the first quadrant of the unit circle.

Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.

Topic E: Trigonometric Ratios in Non-Right Triangles

Derive the area formula for any triangle in terms of sine.

Verify algebraically and find missing measures using the Law of Sines.

Verify algebraically and find missing measures using the Law of Cosines.

Use side and angle relationships in right and non-right triangles to solve application problems.

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5. Trigonometry

5.2 Solve Applications: Sine, Cosine and Tangent Ratios.

Learning Objectives

By the end of this section it is expected that you will be able to

• Find missing side of a right triangle using sine, cosine, or tangent ratios
• Find missing angle of a right triangle using sine, cosine, or tangent ratios
• Solve applications using right angle trigonometry

Sine, Cosine, and Tangent Ratios

We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use θ (theta) to represent it.

The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.

In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.

We labeled  the sides with a lowercase letter to match the uppercase letter of the opposite vertex.

c is hypotenuse

a is opposite

b is adjacent

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

y is hypotenuse

z is opposite

x is adjacent

Trigonometric Ratios

Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.

Let us refer to Figure 1 and define the three basic trigonometric ratios as:

Three Basic Trigonometric Ratios

Where θ is the measure of a reference angle measured in degrees.

Very often we use the abbreviations for sine, cosine, and tangent ratios.

Some people remember the definition of the trigonometric ratios as SOH CAH TOA.

For the given triangle find the sine, cosine and tangent ratio.

For the given triangle find the sine cosine and tangent ratio.

When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.

For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

We have two possible reference angles: R an S.

Using the definitions, the trigonometric ratios for angle R are:

Using the definitions, the trigonometric ratios for angle S:

For the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.

Now, let us use a scientific calculator to find the trigonometric ratios.  Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.

Using a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.

Make sure your calculator is in Degree Mode.

a)  Using a calculator find that sin 30° = 0.5

b)  Using a calculator find that cos 45° = 0.7071 Rounded to 4 decimal places.

c)  Using a calculator find that tan 60° = 1.7321 Rounded to 4 decimal places.

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a)  sin 60°

b)  cos 30°

c)  tan 45°

a)   sin 60° = 0.8660

b)   cos 30° = 0.8660

c)   tan 45° = 1

Finding Missing Sides of a Right Triangle

In this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.

HOW TO: Solve Trigonometry Applications

• Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
• Identify what we are looking for.
• Label what we are looking for by choosing a variable to represent it.
• Find the required trigonometric ratio.
• Solve the ratio using good algebra techniques.
• Check the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.
• Answer the question with a complete sentence.

In the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.

 Find the missing sides. Round  your final answer to two decimal places

 Find the missing sides. Round  your final answer to one decimal place.

Find the hypotenuse. Round your final answer to one decimal place.

Finding Missing Angles of a Right Triangle

Sometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin -1 , cos -1 , and tan -1 are on your scientific calculator.

Find the angles. Round your final answer to one decimal place.

a)  sin A = 0.5

b)  cos B = 0.9735

c)  tan C = 2.89358

Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key

a)  A = sin -1 0.5

b)  B = cos -1 0.9735

c) C = tan -1 2.89358

a)  sin X = 1

b)  cos Y = 0.375

c)  tan Z = 1.676767

In the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.

Find the missing angle X. Round your final answer to one decimal place.

Find the missing angle A. Round your final answer to one decimal place.

Find the missing angle C. Round your final answer to one decimal place.

Solving a Right Triangle

From the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.

Solve the right triangle. Round your final answer to one decimal place.

Since the sum of angles in any triangle is 180°, the measure of angle B can be easy calculated.

We solved the right triangle

Solve the right triangle. Round to two decimal places.

The missing angle F = 180° – 90° – 26.39° = 63.64°

Solve the right triangle. Round to one decimal place.

Solve Applications Using Trigonometric Ratios

In the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let’s use the trigonometric ratios to solve real-life  problems.

Many applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.

The angle of elevation is an angle between the horizontal line (ground) and the observer’s line of sight.

The angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer’s line of sight.

James is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78° angle. How tall is the Harbour Centre?

Marta is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62° angle. How tall is the building?

43.3 metres

Thomas is standing at the top of the building that is 45 metres high and looks at her friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?

Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?

Key Concepts

• Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label  the given parts.
• Check the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.

5.2 Exercise Set

Label the sides of the triangle.

• If the reference angle in Question 1 is B, Find the adjacent ?
• If the reference angle in Question 2 is Z, find the opposite?

Use your calculator to find the given ratios. Round to four decimal places if necessary:

For the given triangles, find the sine, cosine and tangent of the θ.

For the given triangles, find the missing side. Round it to one decimal place.

For the given triangles, find the missing sides. Round it to one decimal place.

Solve the triangles. Round to one decimal place.

• Kim stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48° angle. How tall is the tree?
• A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52°. How tall is the tree?
• A ladder that is 15 feet is leaning against a house and makes a 45° angle with the ground. How far is the base of the ladder from the house?
• Roxanne is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38°. How high is her kite above the ground?
• Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?
• An airplane takes off from the ground at the angle of 25°. If the airplane traveled 200 kilometres, how high above the ground is it?

• y = 19.3, z = 8.2

Attribution:

This chapter has been adapted from “Solve Applications: Sine, Cosine and Tangent Ratios” in Introductory Algebra  by Izabela Mazur, which is under a CC BY 4.0 Licence . See the Adaptation Statement for more information.

Business/Technical Mathematics Copyright © 2021 by Izabela Mazur and Kim Moshenko is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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