Table of Contents 5. Right Triangle Trigonometry

Right Triangle Trigonometry Essential Question – How can right triangles help solve real world applications?

The Pythagorean theorem In a rt Δ the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c2 = a2+b2 c a __ b __

Example x2=72+242 x2=49+576 x2=625 x x=25 7 __ __ 24

Example __ __ x 12

3 basic trig ratios Sine (sin) Cosine (cos) Tangent (tan) Sin  Cos 

Ex: Find sin, cos, & tan of A & B. sin A= 12/13 cos A= 5/13 tan A= 12/5 12 C B B sin B= 5/13 cos B= 12/13 tan B= 5/12 5 13 A

Inverse Trig Functions Cosecant is the inverse of sin Secant is the inverse of cos Cotangent is the inverse of tan

Example: Six Trig Ratios Given that , calculate the other trigonometric functions for  . 4 3 5  Step 1: Draw a right triangle and find third side using Pythagorean theorem. Step 2: Find the other ratios using formulas. sin  = csc  = cos  = sec  = tan  = cot  = Example: Six Trig Ratios

More examples Given that sin = 7/25, sketch the triangle and find the third side. Then find cos Given that tan = ¾, sketch the triangle and find the third side. Then find sin Given that tan = 4/5, what is cot ?

Given a point, find all trig functions 1. Draw right triangle 2. Label theta 3. Label sides 4. Use Pythagorean theorem to find missing side 5. Find all 6 functions

Example Given the point (-4,10) find the values of the six trig function of the angle. 1. Plot point (-4,10) 2. Draw rt triangle 3. Label angle and sides 10.8 10 4. Use Pyt. Th. to find 3rd side. -4 5. Find trig functions

Example Given the point (-5,-2) find the values of the six trig function of the angle. 1. Plot point 2. Draw rt triangle 3. Label angle and sides -5 4. Use Pyt. Th. to find 3rd side. -2 (-5,-2) 5. Find trig functions

Last type of problem You are given a trig ratio It can be in one of two quadrants Therefore you have to be given another piece of information to determine which quadrant it is in

Always Study Trig Carefully Sin y values Cos x values Tan sin/cos Sin + Cos - Tan - Sin + Cos + Tan + Where are these positive? Always Sin All Study Sin - Cos - Tan + Sin - Cos + Tan - Trig Carefully Tan Cos

Steps 1. Find what quadrant the triangle is in 2. Draw right triangle, only sides will be negative, hypotenuse will never be negative 3. Use Pythagorean theorem to find 3rd side 4. Find other trig functions

Example Given that cos θ = 8/17 and tan θ < 0, find all six trig functions. 8 θ -15 17 Triangle is in 4th quadrant because that is where cos is positive and tan is negative