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SECTION 14-4 Right Triangles and Function Values Slide 14-4-1

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RIGHT TRIANGLES AND FUNCTION VALUES Right Triangle Side Ratios Cofunction Identities Trigonometric Function Values of Special Angles Reference Angles Slide 14-4-2

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RIGHT TRIANGLE SIDE RATIOS Slide 14-4-3 The next slide shows an acute angle A in standard position. The definitions of the trigonometric function values of angle A require x, y, and r. x and y are the lengths of the two legs of the right triangle ABC, and r is the length of the hypotenuse. The functions of trigonometry can be adapted to describe the ratios of these sides.

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RIGHT TRIANGLE SIDE RATIOS Slide 14-4-4 B y x x y r (x, y) AC The side of length y is called the side opposite angle A, and the side of length x is called the side adjacent to angle A.

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RIGHT TRIANGLE SIDE RATIOS Slide 14-4-5 The lengths of these sides can be used to replace x and y in the definitions of the trigonometric functions, with r replaced by the length of the hypotenuse.

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RIGHT TRIANGLE-BASED DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Slide 14-4-6 For any acute angle A in standard position, more

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RIGHT TRIANGLE-BASED DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Slide 14-4-7

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EXAMPLE: FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE Slide 14-4-8 A B C 5 13 12 Find the values of the trigonometric functions for angle A in the right triangle. Solution

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COFUNCTION IDENTITIES Slide 14-4-9 A B C a b c

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COFUNCTION IDENTITIES Slide 14-4-10 Because C = 90°, A and B are complementary angles. Because A and B are complementary angles and sin A = cos B, the functions sine and cosine are called cofunctions. Also, tangent and cotangent are cofunctions, as are secant and cosecant. And because A and B are complementary angles, we have B = 90° – A. This leads to sin A = cos B = cos(90° – A). The rest of the cofunction identities are on the next slide.

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COFUNCTION IDENTITIES Slide 14-4-11 For any acute angle A,

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EXAMPLE: WRITING FUNCTIONS IN TERMS OF COFUNCTIONS Slide 14-4-12 Write each of the following in terms of cofunctions. a) cos 48°b) tan 33°c) sec 81° a) sin 42° b) cot 57° c) csc 9° Solution

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TRIGONOMETRIC FUNCTION VALUES OF SPECIAL ANGLES Slide 14-4-13 Certain special angles, such as 30°, 45°, and 60°, occur so often in applications of trigonometry that they deserve special study. The exact trigonometric function values of these angles, found by the properties of geometry and the Pythagorean theorem, are summarized on the next slide.

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TRIGONOMETRIC FUNCTION VALUES OF SPECIAL ANGLES Slide 14-4-14 30° 2 45° 11 60° 2

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REFERENCE ANGLES Slide 14-4-15 Associated with every nonquadrantal angle in standard position is a positive acute angle called its reference angle. A reference angle for an angle written is the positive acute angle made by the terminal side of angle and the x-axis. x x yy

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EXAMPLE: REFERENCE ANGLES Slide 14-4-16 Find the reference angle for 232° Solution x y 232° 52°

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EXAMPLE: REFERENCE ANGLES Slide 14-4-17 Find the reference angle for 1020° Solution x y 300° 60° Find a coterminal angle between 0° and 360°: 1020° – 2(360°) = 300°

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REFERENCE ANGLES, WHERE Slide 14-4-18 x x y y x y x y Q III Q IIQ I Q IV

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FINDING TRIGONOMETRIC FUNCTION VALUES FOR ANY NONQUADRANTAL ANGLE Slide 14-4-19 Step 1If > 360°, or if < 0°, find a coterminal angle by adding or subtracting 360° as many times as needed to obtain an angle greater than 0° but less than 360°. Step 2Find the reference angle Step 3Find the necessary values of the trigonometric functions for the reference angle

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FINDING TRIGONOMETRIC FUNCTION VALUES FOR ANY NONQUADRANTAL ANGLE Slide 14-4-20 Step 4Determine the correct signs for the values found in step 3. This result gives the values of the trigonometric functions for angle

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EXAMPLE: FINDING TRIGONOMETRIC FUNCTION VALUES USING A REFERENCE ANGLE Slide 14-4-21 Use a reference angle to find the exact value of cos 495°. Solution Find a coterminal angle between 0° and 360°: 495° – 360° = 135° In quadrant II, so cosine is negative.

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