Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231

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1 1 Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231

2 What is Trigonometry? 2 It is defined as the study of triangles and the relationships between their sides and the angles between these sides. We will focus specifically on RIGHT TRIANGLES The relationships between sides and angles will be expressed as a RATIO of 2 values The key to solving any trigonometry problem is to IDENTIFY WHICH ANGLE IS BEING USED. NEVER USE THE 90 ANGLE!

3 Identifying Sides of the Right Triangle 3 Recall that the side opposite of the 90 degree angle is called the HYPOTENUSE. The legs of the right triangle (the remaining two sides) are called the OPPOSITE SIDE and the ADJACENT SIDE. Their location is always based off of which angle is being used. The opposite side is always located ACROSS from the angle being used. The adjacent side is always located NEXT TO the angle being used (but is not the hypotenuse).

4 Example Identify the opposite side, adjacent side, and hypotenuse in the triangle below if the angle of interest is angle A. 4 Now do it for angle C.

5 Setting Up the Trigonometric Ratios The three basic trigonometric ratios are SINE (SIN), COSINE (COS), and TANGENT (TAN) and they are formed by making ratios of two of the three sides of the right triangle 5 The sine ratio is formed as sin θ = opposite, where θ represents the angle of interest. hypotenuse The cosine ratio is formed as cos θ = adjacent hypotenuse The tangent ratio is formed as tan θ = opposite adjacent Use the phrase SOH CAH TOA to help you remember how each ratio is formed.

6 Example 1 Page 235 Find the sine, cosine, and tangent ratios for A and B in ΔABC. Convert the ratios to decimal equivalents. (Hint: First find the length of the hypotenuse using the Pythagorean Theorem. ) 6

7 More Trigonometric Ratios The three trigonometric ratios (sin, cos, and tan) also have reciprocals. Recall that a reciprocal is a value that when multiplied by the original value, the product is 1. 7 The reciprocal of sine is cosecant or csc csc θ = hypotenuse opposite The reciprocal of cosine is secant or sec sec θ = hypotenuse adjacent The reciprocal of tangent is cotangent or cot cot θ = adjacent opposite

8 Example 2 Page 236 Given the triangle below, set up the three six trigonometric ratios of sine, cosine, tangent, cosecant, secant, and cotangent for the angle given. Compare these ratios to the trigonometric functions using your calculator. 8

9 Example (Not in book) Given the value of 1 trig function find the values of the other 5 trig functions of θ. 9 cscθ = 34 5

10 Example 3 Page 236 A right triangle has a hypotenuse of 5 and a side length of 2. Find the angle measurements and the unknown side length. Find the sine, cosine, and tangent for both angles. Without drawing another triangle, compare the trigonometric ratios of ΔABC with those of a triangle that has been dilated by a factor of k = What do you notice about the sin and cos of the 2 different angles?

11 Complementary Angles: (Page 252) Recall that complementary angles are two angles that add up to 90 degrees. In any right triangle, the two acute angles are always complementary. That means that if one of the acute angles has a measure of x, then the other acute angle may be found by finding 90-x. Example: Find the complement of angle x. a) x = b) x = 37

12 Sin and Cos as Cofunctions Complementary angles forms the basis of the sin-cos cofunction identity. The sin-cos cofunction identity can be written as: sinθ = cos 90 θ cosθ = sin(90 θ) 12 This means that the sine of one acute angle can be used to find the cosine of its complementary angle or the cosine of one acute angle can be used to find the sine of its complementary angle.

Example 1 Page 249 Find sin 28 o if cos 62 o is approximately 0.469.

13 Example 1 Page 249 Find sin 28 o if cos 62 o is approximately Example 2 Page 250 Complete the table below using the sine and cosine identities

14 Example 3 Page 250 Find a value of θ for which sin θ = cos 15º is true. 14 Example 4 Page 250 Complete the table below using the sine and cosine identities

15 15 Assignment 10.1 Problems # s 1 12 from the packet AND Page 253 (WB 5) # s 1 8

9.1 Use Trigonometry with Right Triangles

9.1 Use Trigonometry with Right Triangles Use the Pythagorean Theorem to find missing lengths in right triangles. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle