Slide 1-1 Chapter 2 Acute Angles and Right Triangle Y. Ath.

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Chapter 2 Acute Angles and Right Triangle

Y. Ath

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Section 2. 1Trigonometric Functions of Acute Angle

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Six Trigonometric Functions using Right Triangle

hypotenuse

opposite)sin(

hypotenuse

adjacent)cos(

adjacent

opposite)tan(

opposite

hypotenuse

)sin(

1)csc(

adjacent

hypotenuse

)cos(

1)sec(

opposite

adjacent

)tan(

1)cot(

SOH-CAH-TOA

1. Sine

2. Cosine

3. Tangent

4. Cosecant

5. Secant

6. Cotangent

Note:

1sinsin

1

4

Find the sine, cosine, and tangent values for angles A and B.

Example 1

85

77ASin

85

36BSin

85

36A Cos

85

77B Cos

36

77ATan

36

77BCot

B CosASin

BSin A Cos

BCot ATan

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Cofunction Identities

)90cos(sin )1( 0 AA

)90sin(cos )2( 0 AA

)90cot(tan )3( 0 AA

)90csc(sec )4( 0 AA

)90sec(csc )5( 0 AA

)90tan(cot )6( 0 AA

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

Example 2 SOLVING EQUATIONS USING COFUNCTION IDENTITIES

(a)

(b)

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Special Triangles

45º-45º-90º (Isosceles Right Triangle) 30º-60º-90º (Equilateral Triangle)

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Section 2.2 Trigonometric Functions of Non-Acute Angles

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Reference Angles

Quad IQuad IQuad IIQuad II

Quad IIIQuad III Quad IVQuad IV

θ’ = θ θ’ = 180° – θ

θ’ = θ – 180° θ’ = 360° – θ

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Example 3 Find the values of the six trigonometric

functions for 210°.

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Finding Trigonometric Function Values For Any Nonquadrantal Angle θ

Step 1 If θ > 360°, or if θ < 0°, find a coterminal angle by adding or subtracting 360° as many times as needed to get an angle greater than 0° but less than 360°.Step 2 Find the reference angle θ′.

Step 3 Find the trigonometric function values for reference angle θ′.

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Finding Trigonometric Function Values For Any Nonquadrantal Angle θ (continued)Step 4 Determine the correct signs

for the values found in Step 3. This gives the values of the trigonometric functions for angle θ.

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Find the exact value of sin (–150°).

Example 4

Step 1. Find a coterminal angle of –150° 000 210360150 cA

Step 2. Find a reference angle.

000 30180210'

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Find the exact value of cot 780°.

Example 5

Step 1. Find a coterminal angle of 780° 00000 607207803602780 cA

Step 2. Find a reference angle.060'

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Evaluate .

Example 6

00202 225tan2135cos345sin

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Example 7

2

3sin and )360 [0, interval in the is if , of valuesall Find 0

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Section 2.3 Finding Trig Function Values Using a Calculator

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5.0cos ,5.0cos if -1

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Example 8 FINDING GRADE RESISTANCE

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Example 8 FINDING GRADE RESISTANCE (cont.)

(a) Calculate F to the nearest 10 lb for a 2500-lb car traveling an uphill grade with θ = 2.5°.

(b) Calculate F to the nearest 10 lb for a 5000-lb truck traveling a downhill grade with θ = –

6.1°.

F is negative because the truck is moving downhill.

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Example 8 FINDING GRADE RESISTANCE (cont.)

(c) Calculate F for θ = 0° and θ = 90°. Do these answers agree with your intuition?

If θ = 0°, then there is level ground and gravity does not cause the vehicle to roll.

If θ = 90°, then the road is vertical and the full weight of the vehicle would be pulled downward by gravity, so F = W.

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Significant Digits

A significant digit is a digit obtained by actual measurement.

Your answer is no more accurate than the least accurate number in your calculation.

The significant digits in the following numbers are identified in color.

408 21.5 18.00 6.700 0.0025 0.09810 7300

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To determine the number of significant digits for answers in applications of angle measure, use the following table.

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Solving Triangles

To solve a triangle means to find the measures of all the angles and sides of the triangle.

Use a to represent the length of the side opposite angle A, b for the length of the side opposite angle B, and so on.

In a right triangle, the letter c is reserved for the hypotenuse.

When solving triangles, a labeled sketch is an important aid.

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Example 9 SOLVING A RIGHT TRIANGLE GIVEN AN ANGLE AND A SIDE

Solve right triangle ABC, if A = 34°30′ and c = 12.7 in.

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Example 11 SOLVING A RIGHT TRIANGLE GIVEN TWO SIDES

Solve right triangle ABC,

if a = 29.43 cm and c = 53.58 cm.

or 33º 19΄

90 33 19 56 41B ' '

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Angles of Elevation or Depression

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Example FINDING A LENGTH GIVEN THE ANGLE OF ELEVATION

Pat Porterfield knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to the top of the flagpole is 26°40′. If her eyes are 5.30 ft above the ground, find the height of the flagpole.

Since Pat’s eyes are 5.30 ft above the ground, the height of the flagpole is 61.8 + 5.30 = 67.1 ft.

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Example 12 FINDING AN ANGLE OF DEPRESSION

From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the angle of depression from the top of the cliff to the base of the lighthouse.

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Example 12 FINDING AN ANGLE OF DEPRESSION (continued)

210tan ,

430B 1 210

so tan430

B

Angle of depre26 s o n si

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Bearing

There are two methods for expressing bearing.

When a single angle is given, such as 164°, it is understood that the bearing is measured in a clockwise direction from due north.

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Example 13 SOLVING A PROBLEM INVOLVING BEARING (METHOD 1)

Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to C.

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Bearing

The second method for expressing bearing starts with a north-south line and uses an acute angle to show the direction, either east or west, from this line.

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Example 14 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2)

A ship leaves port and sails on a bearing of N 47º E for 3.5 hr. It then turns and sails on a bearing of S 43º E for 4.0 hr. If the ship’s rate of speed is 22 knots (nautical miles per hour), find the distance that the ship is from port.

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Example 14 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2) (cont.)

22 3.5 77 nautical milesb 22 4.0 88 nautical milesa

2 2 2a b c Now find c, the distance from port at point A to the ship at point B.

2 2 288 77 c 2 288 77 120 nautical mic

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Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE

The subtense bar method is a method that surveyors use to determine a small distance d between two points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. Angle θ is measured, then the distance d can be determined.

(a) Find d with θ = 1°23′12″ and b = 2.0000 cm.

From the figure, we have

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Let b = 2. Convert θ to decimal degrees:

Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued)

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Since θ is 1″ larger, θ = 1°23′13″ ≈ 1.386944º.

(b) How much change would there be in the value of d if θ were measured 1″ larger?

Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued)

The difference is

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A surveyor is standing 115 feet from the base of the Washington

Monument. The surveyor measures the angle of elevation to the

top of the monument as 78.3. How tall is the Washington Monument?

Solution:

where x = 115 and y is the height of the monument. So, the height of the

Washington Monument is

y = x tan 78.3 115(4.82882) 555 feet.

Example 16

Slide 1-1 Chapter 2 Acute Angles and Right Triangle Y. Ath.

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