Section 14-4 Right Triangles and Function Values

Right Triangles and Function Values Right Triangle Side Ratios Cofunction Identities Trigonometric Function Values of Special Angles Reference Angles

Right Triangle Side Ratios 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.

Right Triangle Side Ratios 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. y B (x, y) r y A C x x

Right Triangle Side Ratios 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.

Right Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position, more

Right Triangle-Based Definitions of Trigonometric Functions

Example: Finding Trigonometric Function Values of an Acute Angle Find the values of the trigonometric functions for angle A in the right triangle. B 13 5 Solution A C 12

Cofunction Identities B c a A C b

Cofunction Identities 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.

Cofunction Identities For any acute angle A,

Example: Writing Functions in Terms of Cofunctions Write each of the following in terms of cofunctions. a) cos 48° b) tan 33° c) sec 81° Solution a) sin 42° b) cot 57° c) csc 9°

Trigonometric Function Values of Special Angles 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.

Trigonometric Function Values of Special Angles 30° 2 45° 1 60°

Reference Angles 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. y y x x

Example: Reference Angles Find the reference angle for 232° Solution y 232° x 52°

Example: Reference Angles Find the reference angle for 1020° Solution Find a coterminal angle between 0° and 360°: 1020° – 2(360°) = 300° y 300° x 60°

Reference Angles, Where Q II Q I y y x x Q III Q IV y y x x

Finding Trigonometric Function Values for Any Nonquadrantal Angle Step 1 If > 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 2 Find the reference angle Step 3 Find the necessary values of the trigonometric functions for the reference angle

Finding Trigonometric Function Values for Any Nonquadrantal Angle Step 4 Determine the correct signs for the values found in step 3. This result gives the values of the trigonometric functions for angle

Example: Finding Trigonometric Function Values Using a Reference Angle 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.