Right Triangle Trigonometry Trigonometry is based upon ratios of the sides of right triangles. The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. θ opp hyp adj The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle.

A A The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90 o. Right Triangle Trigonometry

S O H C A H T O A The trigonometric functions are: sine, cosine, tangent, cotangent, secant, and cosecant. opp adj hyp θ sin  = cos  = tan  = csc  = sec  = cot  = opp hyp adj hyp adj opp adj Trigonometric Ratios

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. Find angle C a)Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters. 14 cm 6 cm C 1.

C = cos -1 (0.4286) C = 64.6 o 14 cm 6 cm C 1. h a We have been given the adjacent and hypotenuse so we use COSINE: Cos A = Cos C = Cos C =

Find angle x2. 8 cm 3 cm x a o Given adj and opp need to use tan: Tan A = x = tan -1 (2.6667) x = 69.4 o Tan A = Tan x = Tan x =

Cos 30 x 7 = k 6.1 cm = k 7 cm k 30 o 3. We have been given the adj and hyp so we use COSINE: Cos A = Cos 30 = Finding a side from a triangle

Tan 50 x 4 = r 4.8 cm = r 4 cm r 50 o 4. Tan A = Tan 50 = We have been given the opp and adj so we use TAN: Tan A =

45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. x√2 45° Hypotenuse = √2 * leg

30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. x√3 60° 30° Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

Ex. 1: Finding the hypotenuse in a 45°-45°- 90° Triangle Find the value of x By the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°- 90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. 33 x 45°

Ex. 1: Finding the hypotenuse in a 45°-45°- 90° Triangle Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 33 x 45° 45°-45°-90° Triangle Theorem Substitute values Simplify

Ex. 3: Finding side lengths in a 30°-60°-90° Triangle Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. 30° 60°

Ex. 3: Side lengths in a 30°-60°-90° Triangle Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem 5 √3√3 √3 s √3√3 = 5 √3√3 s= 5 √3√3 s= √3 √3√3 5√35√3 3 s= Substitute values Divide each side by √3 Simplify Multiply numerator and denominator by √3 Simplify 30° 60°

The length t of the hypotenuse is twice the length s of the shorter leg. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem t2∙2∙ 5√ 3 3 = Substitute values Simplify 30° 60° t 10√ 3 3 =