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Trigonometric Functions

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1.1 Angles

1.2 Trigonometric Functions

1.3 Using the Definitions of the Trigonometric Functions

Trigonometric Functions1

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Angles1.1Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles

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For an angle measuring 55°, find the measure of its complement and its supplement.

1.1 Example 1 Finding the Complement and the Supplement of an Angle

Complement: 90° − 55° = 35°

Supplement: 180° − 55° = 125°

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Find the measure of each angle.

1.1 Example 2(a) Finding Measures of Complementary and Supplementary Angles

The two angles form a right angle, so they are complements.

The measures of the two angles are

and

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Find the measure of each angle.

1.1 Example 2(b) Finding Measures of Complementary and Supplementary Angles (page 3)

The two angles form a straight angle, so they are supplements.

The measures of the two angles are

and

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Perform each calculation.

1.1 Example 3 Calculating with Degrees, Minutes, and Seconds

(a)

(b)

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1.1 Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds

(a) Convert 105°20′32″ to decimal degrees.

(b) Convert 85.263° to degrees, minutes, and seconds.

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Find the angles of least possible positive measure coterminal with each angle.

1.1 Example 5 Finding Measures of Coterminal Angles

(a) 1106°

(b) –150°

Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°.

An angle of 1106° is coterminal with an angle of 26°.

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

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1.1 Example 5 Finding Measures of Coterminal Angles

(c) –603°

An angle of –603° is coterminal with an angle of 117°.

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1.1 Example 6 Analyzing the Revolutions of a CD Player

A wheel makes 270 revolutions per minute. Through how many degrees will a point on the edge of the wheel move in 5 sec?

The wheel makes 270 revolutions in one minute or revolutions per second.

In five seconds, the wheel makesrevolutions.

Each revolution is 360°, so a point on the edge of the wheel will move

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Trigonometric Functions1.2Trigonometric Functions ▪ Quadrantal Angles

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The terminal side of an angle θ in standard position passes through the point (12, 5). Find the values of the six trigonometric functions of angle θ.

1.2 Example 1 Finding Function Values of an Angle

x = 12 and y = 5.

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The terminal side of an angle θ in standard position passes through the point (8, –6). Find the values of the six trigonometric functions of angle θ.

1.2 Example 2 Finding Function Values of an Angle

x = 8 and y = –6.

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1.2 Example 2 Finding Function Values of an Angle

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Find the values of the six trigonometric functions of angle θ in standard position, if the terminal side of θ is defined by 3x – 2y = 0, x ≤ 0.

1.2 Example 3 Finding Function Values of an Angle

Since x ≤ 0, the graph of the line 3x – 2y = 0 is shown to the left of the y-axis.

Find a point on the line:Let x = –2. Then

A point on the line is (–2, –3).

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1.2 Example 3 Finding Function Values of an Angle

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Find the values of the six trigonometric functions of a 360° angle.

1.2 Example 4(a) Finding Function Values of Quadrantal Angles

The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2.

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Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5).

1.2 Example 4(b) Finding Function Values of Quadrantal Angles

x = 0 and y = –5 and r = 5.

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Using the Definitions of the Trigonometric Functions1.3Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities

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Find each function value.

1.3 Example 1 Using the Reciprocal Identities

(a) tan θ, given that cot θ = 4.

(b) sec θ, given that

tan θ is the reciprocal of cot θ.

sec θ is the reciprocal of cos θ.

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Determine the signs of the trigonometric functions of an angle in standard position with the given measure.

1.3 Example 2 Finding Function Values of an Angle

(a) 54° (b) 260° (c) –60°

(a) A 54º angle in standard position lies in quadrant I, so all its trigonometric functions are positive.

(b) A 260º angle in standard position lies in quadrant III, so its sine, cosine, secant, and cosecant are negative, while its tangent and cotangent are positive.

(c) A –60º angle in standard position lies in quadrant IV, so cosine and secant are positive, while its sine, cosecant, tangent, and cotangent are negative.

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Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions.

1.3 Example 3 Identifying the Quadrant of an Angle

(a) tan θ > 0, csc θ < 0

(b) sin θ > 0, csc θ > 0

tan θ > 0 in quadrants I and III, while csc θ < 0 in quadrants III and IV. Both conditions are met only in quadrant III.

sin θ > 0 in quadrants I and II, as is csc θ. Both conditions are met in quadrants I and II.

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Decide whether each statement is possible or impossible.

1.3 Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function

(a) cot θ = –.999 (b) cos θ = –1.7 (c) csc θ = 0

(a) cot θ = –.999 is possible because the range of cot θ is

(b) cos θ = –1.7 is impossible because the range of cos θ is [–1, 1].

(c) csc θ = 0 is impossible because the range of csc θ is

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Angle θ lies in quadrant III, and Find the values of the other five trigonometric functions.

1.3 Example 5 Finding All Function Values Given One Value and the Quadrant

Since and θ lies in quadrant III, then x = –5 and y = –8.

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1.3 Example 5 Finding All Function Values Given One Value and the Quadrant

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1.3 Example 6 Finding Other Function Values Given One Value and the Quadrant

Find cos θ and tan θ given that sin θ andcos θ > 0.

Reject the negative root.

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1.3 Example 6 Finding Other Function Values Given One Value and the Quadrant (cont.)

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1.3 Example 7 Finding Other Function Values Given One Value and the Quadrant

Find cot θ and csc θ given that cos θ andθ is in quadrant II.

Since θ is in quadrant II, cot θ < 0 and csc θ > 0.

Reject the negative root:

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1.3 Example 7 Finding Other Function Values Given One Value and the Quadrant (cont.)