2 Acute Angles and Right Triangles © 2008 Pearson Addison-Wesley. All rights reserved

2

Acute Angles and Right Triangles

© 2008 Pearson Addison-Wesley.All rights reserved

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2-2

2.1 Trigonometric Functions of Acute Angles

2.2 Trigonometric Functions of Non-Acute Angles

2.3 Finding Trigonometric Function Values Using a Calculator

2.4 Solving Right Triangles

2.5Further Applications of Right Triangles

Acute Angles and Right Triangles2


Trigonometric Functions of Acute Angles

2.1

Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles


Find the sine, cosine, and tangent values for angles D and E in the figure.

2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (page 51)


Find the sine, cosine, and tangent values for angles D and E in the figure.

2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (cont.)


Write each function in terms of its cofunction.

2.1 Example 2 Writing Functions in Terms of Cofunctions

(page 51)

(a) sin 9° = cos (90° – 9°) = cos 81°

(b) cot 76° = tan (90° – 76°) = tan 14°

(c) csc 45° = sec (90° – 45°) = sec 45°


Find one solution for the equation. Assume all angles involved are acute angles.

2.1 Example 3(a) Solving Equations Using the Cofunction Identities (page 52)

(a)

Since cotangent and tangent are cofunctions, the equation is true if the sum of the angles is 90º.


Find one solution for the equation. Assume all angles involved are acute angles.

2.1 Example 3(b) Solving Equations Using the Cofunction Identities (page 52)

(b)

Since secant and cosecant are cofunctions, the equation is true if the sum of the angles is 90º.


Find the six trigonometric function values for a 30° angle.

2.1 Example 5 Finding Trigonometric Values for 30° (page 54)


2.1 Example 5 Finding Trigonometric Values for 30° (cont.)


Trigonometric Functions of Non-Acute Angles2.2Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles


Find the reference angle for 294°.

2.2 Example 1(a) Finding Reference Angles (page 59)

294 ° lies in quadrant IV.

The reference angle is 360° – 294° = 66°.


Find the reference angle for 883°.

2.2 Example 1(b) Finding Reference Angles (page 59)

Find a coterminal angle between 0° and 360° by dividing 883° by 360°. The quotient is about 2.5.


883° is coterminal with 163°.


Find the values of the six trigonometric functions for 135°.

2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 60)

The reference angle for 135° is 45°.

Choose point P on the terminal side of the angle. The coordinates of P are (1, –1).


2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 60)


2.2 Example 3(a) Finding Trigonometric Function Values Using Reference Angles (page 61)

Find the exact value of sin(–150°).

An angle of –150° is coterminal with an angle of –150° + 360° = 210°.


Since an angle of –150° lies in quadrant III, its sine is negative.


2.2 Example 3(b) Finding Trigonometric Function Values Using Reference Angles (page 61)

Find the exact value of cot(780°).

An angle of 780° is coterminal with an angle of 780° – 2 ∙ 360° = 60°.

The reference angle is 60°.

Since an angle of 780° lies in quadrant I, its cotangent is positive.


2.2 Example 4 Evaluating an Expression with Function Values of Special Angles (page 62)


2.2 Example 5 Using Coterminal Angles to Find Function Values (page 62)

Evaluate each function by first expressing the function in terms of an angle between 0° and 360°.

(a)

(b)


2.2 Example 6 Finding Angle Measures Given an Interval and a Function Value (page 63)

Find all values of θ, if θ is in the interval [0°, 360°) and

sin θ =

Since sin θ is negative, θ must lie in quadrants III or IV.

The absolute value of sin θ is so the reference angle is 60°.

The angle in quadrant III is 60° + 180° = 240°.

The angle in quadrant IV is 360° – 60° = 300°.


Finding Trigonometric Functions Using a Calculator2.3Finding Function Values Using a Calculator ▪ Finding Angle Measures Using a Calculator


Approximate the value of each expression.

2.3 Example 1 Finding Function Values with a Calculator

(page 67)


2.3 Example 1 Finding Function Values with a Calculator

(cont.)


2.3 Example 2 Using Inverse Trigonometric Functions to Find Angles (page 67)

Use a calculator to find an angle θ in the interval[0°, 90°] that satisfies each condition.


2.3 Example 3 Finding Grade Resistance (page 68)

The force F in pounds when an automobile travels uphill or downhill on a highway is called grade resistance and is modeled by the equation , where θ is the grade and W is the weight of the automobile. If the automobile is moving uphill, then θ > 0°; if it is moving downhill, then θ < 0°.

(a) Calculate F to the nearest 10 pounds for a 5500-lb car traveling an uphill grade with θ = 3.9°.


2.3 Example 3 Finding Grade Resistance (cont.)

(b) Calculate F to the nearest 10 pounds for a 2800-lb car traveling a downhill grade with θ = –4.8°.

(c) A 2400-lb car traveling uphill has a grade resistance of 288 pounds. What is the angle of the grade?


Solving Right Triangles2.4Significant Digits ▪ Solving Triangles ▪ Angles of Elevation or Depression


Solve right triangle ABC, if B = 28°40′ and a = 25.3 cm.

2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (page 75)


2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.)

Three significant digits







Solve right triangle ABC, if a = 44.25 cm and b = 55.87 cm.

2.4 Example 2 Solving a Right Triangle Given Two Sides

(page 76)

Use the Pythagorean theorem to find c:

Four significant digits



(cont.)



(cont.)


The angle of depression from the top of a tree to a point on the ground 15.5 m from the base of the tree is 60.4°. Find the height of the tree.

2.4 Example 3 Finding a Length When the Angle of Elevation is Known (page 77)

The measure of equals the measure of the angle of depression because the two angles are alternate interior angles, so


2.4 Example 3 Finding a Length When the Angle of Elevation is Known (cont.)

The tree is about 27.3 m tall.



The length of a shadow of a flagpole 55.20 ft tall is 27.65 ft. Find the angle of elevation of the sun.

2.4 Example 4 Finding a Length When the Angle of Elevation is Known (page 78)

The angle of elevation of the sun is 63.39°.

Four significant digits


Further Applications of Right Triangles2.5Bearing ▪ Further Applications


Radar stations A and B are on an east-west line, 8.6 km apart. Station A detects a plane at C, on a bearing of 53°. Stations B simultaneously detects the same plane, on a bearing of 323°. Find the distance from B to C.

2.5 Example 1 Solving Problem Involving Bearing (First Method) (page 83)

A line drawn due north is perpendicular to an east-west line, so right angles are formed at A and B, and angles CAB and CBA can be found as shown in the figure.

Angle C is a right angle because angles CAB and CBA are complementary.


Find distance a by using the cosine function for angle A.

2.5 Example 1 Solving Problem Involving Bearing (First Method) (cont.)

The distance from B to C is about 5.2 km.


The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph take 1.8 hours to go from A to B. Find the distance from B to C.

2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 84)

The information in the example gives and



The sum of the measures of angles EAB and FBA is 180º because they are interior angles on the same side of a transversal.

98°

34°56°

90°



It takes 1.8 hours at 350 mph to fly from A to B, so

98°

34°56°

90°

630 mi

The distance from B to C is about 350 miles.


Marla needs to find the height of a building. From a given point on the ground, she finds that the angle of elevation to the top of the building is 74.2°. She then walks back 35 feet. From the second point, the angle of elevation to the top of the building is 51.8°. Find the height of the building.

2.5 Example 4 Solving a Problem Involving Angles of Elevation (page 85)


There are two unknowns, the distance from the base of the building, x, and the height of the building, h.

2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)

In triangle ABC

In triangle BCD


Set the two expressions for h equal and solve for x.


Since h = x tan 74.2°, substitute the expression for x to find h.

The building is about 69 feet tall.


Graphing Calculator Solution


Superimpose coordinate axes on the figure with D at the origin.

The coordinates of A are (35, 0).

The tangent of the angle between the x-axis and the graph of a line with equation y = mx + b is the slope of the line. For line DB, m = tan 51.8º.



Since b = 0, the equation of line DB is

The equation of line AB is

Use the coordinates of A and the point-slope form to find the equation of AB:



Graph y1 and y2, then find the point of intersection. The y-coordinate gives the height, h.

The building is about 69 feet tall.

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2 Acute Angles and Right Triangles © 2008 Pearson Addison-Wesley. All rights reserved