Table of Contents 3 Right Triangle Trigonometry Right

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. c 2 = a 2+b 2 c a __ __ b

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!)

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

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

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

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