Table of Contents 3. 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.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. __ a b c c 2 = a 2 +b 2

Pythagorean Theorem We are going to rewrite the Pythagorean theorem for the special right triangles in a unit circle x 2 + y 2 = 1 We can also rewrite this using the sin/cos relationship on the unit circle cos 2 + sin 2 = 1 This is called a Pythagorean trig identity (more on this later!)

Given a trig function (assuming 1 st quadrant), find other 5 trig functions Step 2: Find the other ratios using what we learned about trig ratios Example: Six Trig Ratios Step 1: Use the Pythagorean trig identity to find sin or cos 1 st type of problem

Given that, calculate the other trigonometric functions for . Step 2: Find the other ratios using formulas. sin  = tan  = sec  = cos  = cot  = csc  = Example: Six Trig Ratios Step 1: Use Pythagorean trig identity to find cos cos  =

More examples Given that sin = 7/25, find cos Given that tan = ¾, find sin

2 nd type - 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. (-4,10) 1. Plot point 2. Draw rt triangle 3. Label angle and sides Use Pyt. Th. to find 3 rd side. 5. Find trig functions

Example Given the point (-5,-2) find the values of the six trig function of the angle. (-5,-2) 1. Plot point 2. Draw rt triangle 3. Label angle and sides Use Pyt. Th. to find 3 rd side. 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 + Cos + Tan + Where are these positive? Sin + Cos - Tan - Sin - Cos - Tan + Sin - Cos + Tan - AllSin TanCos Always Study Trig Carefully Sin y values Cos x values Tan sin/cos

Steps 1. Find what quadrant the triangle is in 2. Use Pythagorean trig identity to find sin or cos 3. Find other trig functions remembering which are positive and negative based on the quadrant

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

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