WARM UP How many degrees are in a right angle? 90° Use the Pythagorean theorem to find the length of side c. 90° 8 + 6 = C c 64 + 36 = 100 6 C = 100 8 C = 10

TRIGONOMETRIC FUNCTIONS IN TRIANGLES

OBJECTIVES Find the sine, cosine and tangent for an angle of a right triangle Find the lengths of sides in special triangles Find the six trigonometric function values for an angle given one of the function values

TRIGONOMETRIC RATIOS The word trigonometry means “triangle measurement.” the Greeks and Hindus saw it mainly as a tool for astronomy. There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. In a right triangle, the side opposite the right angle is the hypotenuse. In the triangle, the hypotenuse has length c, the side opposite the angle θ (theta) has length a and the side adjacent to θ has length b. c a θ b

DEFINITION The ratio depends on θ, and is a function of θ. This function is the sine function. Sine function: sin θ = length of the side opposite θ length of the hypotenuse Cosine function: cos θ = length of the side adjacent to θ Tangent function: tan θ = length of the side opposite θ___ length of the side adjacent to θ

EXAMPLE 1 Because all right triangles with an angle of measure θ are similar, function values depend only on the size of the angle, not the size of the triangle. In this triangle, find the sin θ, cos θ and tan θ. sin θ = side opposite θ 3 hypotenuse 5 5 3 cos θ = side adjacent θ 4 hypotenuse 5 θ tan θ = side opposite θ 3 side adjacent θ 4 4

TRY THIS… In this triangle, find the sin θ, cos θ and tan θ. sin θ = side opposite θ 4 hypotenuse 5 5 3 cos θ = side adjacent θ 3 hypotenuse 5 tan θ = side opposite θ 4 side adjacent θ 3 4

SPECIAL ANGLES Our knowledge of triangles enables us to determine trigonometric function values fro certain angles. First recall the Pythagorean theorem. It says that in any right triangle , where c is the length of the hypotenuse. c b a In a 45° right triangle the legs are the same length. Consider such a triangle whose legs have length 1. Then its hypotenuse has length c. or or

SPECIAL ANGLES Such a triangle is shown below. From this diagram we can easily determine the trigonometric function values for 45°. √2 1 1 Next we consider an equilateral triangle with sides of length 2. if we bisect one angle, we obtain a right triangle that has a hypotenuse of length 2 and a leg of length 1. the other leg has a length of a, given by the Pythagorean theorem as follows: 2 60° 1 30° a 1 2 60° or or

ACUTE ANGLES The acute angles of this triangle have measures of 30° and 60°. We can now determine function values for 30° and 60°. 1 2 30° √3 We can use what we have learned about trigonometry to solve problems.

EXAMPLE 2 In ABC, b = 40 cm and m A = 60°. What is the length of side c? B c Substituting 60° Using cos 60 = 1/2 C A b

TRY THIS… In PQR, q = 12 ft. Use cosine function to find the length of side r Q r P p 45° q R

RECIPROCAL FUNCTIONS We define the three other trigonometric functions by finding the reciprocals of sine, cosine and tangent functions. DEFINITION: The cotangent, secant, and cosecant functions are the respective reciprocal of the tangent, cosine and sine functions. cot θ = length of the side adjacent to θ length of the side opposite θ sec θ = length of the hypotenuse______ length of the side adjacent to θ csc θ = length of the hypotenuse___ length of the side opposite to θ

EXAMPLE 3 Find the cotangent, secant, and cosecant of the angle shown. Approximate to two decimal places. θ cotan θ = side adjacent θ = 3 = 0.75 side opposite θ 4 5 3 sec θ = hypotenuse__ = 5 = 1.67 side adjacent θ 3 csc θ = hypotenuse___ = 5 = 1.25 side opposite θ 4 4

TRY THIS… Approximate cot θ, sec θ and csc θ to two decimal places. cotan θ = 1.33 sec θ = 1.25 5 csc θ = 1.67 3 θ 4

EXAMPLE 4 By using the Pythagorean Theorem we can find all six trigonometric function values of θ when one of the ratios is known. If sine θ = , find the other five trigonometric function values for θ. We know from the definition of the sine function that the ratio is θ side opposite θ is hypotenuse Sin θ=

USING TRIGONOMETRIC FUNCTIONS Let us consider a similar right triangle in which the hypotenuse has length 13 and the side opposite θ has length 12. θ 13 a Choosing the positive square root, since we are finding length. 12 We can use a = 5, b = 12, and c =12 to find all of the ratios in our original triangle.

TRY THIS… If cos θ = , find the other five trigonometric function values for θ. a = 15