Quiz

Warm Up 5.3 Solve for ALL possible solutions 1) cos(x) = 3 2 5) Cot(x) = -1 2) tan(x) = undefined 6) Csc(x) = -2 3) cos(x) = −1 7) Sec(x) = 2 4) cos(x) = - 3 2 8) tan(x) = - 1 3 1 pi/6 11pi/6 2 pi/2 3pi.2 3 pi 4 5pi.6, 7pi.6 5 3pi/4, 7pi/4 6 7pi/6, 11pi/6 7 pi/4, 7pi.4 8 5pi/6, 11pi/6

Ch 5.3 Notes To solve a trig equation, use standard algebraic techniques such as collecting like terms and factoring. Your goal is to isolate the trigonometric function involved in the equation.

EX 1: Solving a Trigonometric Equation 2 sin(x) – 1 = 0 What values of x make this equation true? 2 sin(x) = 1 Add 1 to each side sin(x) = ½ Divide by 2 Where is this true? First let us look on the interval [0, 2π] π 6 and 5π 6

x = π 6 + 2nπ and x = 5π 6 + 2nπ

*Some Possible Moves * Combine Like Terms Factor Foil Get a common denominator / Combine Fractions Rewrite an equation using all sine or all cosine (or all of one trig identity) Square both sides Use a Pythagorean identity Ex: 1 + 𝑡𝑎𝑛 2 (x) = 𝑠𝑒𝑐 2 (x) Rewrite a trig expression using its reciprocal identity Ex: sin(x) = 1 csc⁡(𝑥)

EX 2: Find all solutions of sin(x) + 2 = -sin(x) on the interval [0, 2π] sin(x) + 2 = -sin(x) Original Equation sin(x) + sin(x) = - 2 add sin(x) to and subtract sqrt(2) to each side 2sin(x) = - 2 Combine like terms sin(x) = - √2 2 Divide each side by 2 Where is this true? x = 5π/4 and x = 7π/4

You Try! EX 3: Solve 3 tan²x -1 = 0 Pi/6 + npi and 5pi/6 + npi Remember tangent has a period of π so start by looking on the interval [0, π]

You Try! 1) Find all solutions of 2 𝑠𝑖𝑛 2 (x) – sin(x) -1 = 0 in the interval [0,2π) Check your answer with your partner. Then use your graphing calculator (set in radian mode) to graph the function. Use the zoom and trace features to approximate the x-intercepts. Describe what you and your partner notice in at least 1 complete sentence

EX 3: Solve cot(x)cos²(x) = 2 cot(x) cot(x)cos²(x) = 2 cot(x) Rewrite original cot(x)cos²(x) - 2 cot(x) = 0 Subtract to get trig functions on one side cot(x) [cos²(x) – 2] = 0 Factor out the cot(x) So cot(x) = 0 and cos²(x) – 2=0 Set each equal to 0 cos(x) = ± 2 no answer because ± 2 outside range of cosine function x = π 2 and 3π 2 x = π 2 + nπ only solution

EX 4: Find all solutions of cos(x) + 1 = sin(x) Notice - it is not clear how to rewrite this equation in terms of a single trigonometric function. Stuck? Try squaring both sides and seeing what happens! cos(x) + 1 = sin(x) cos 2 (x) + 2cos(x) + 1 = sin 2 (x) Squared each side cos 2 (x) + 2cos(x) + 1 = 1- cos 2 (x) Use Pythagorean Identity to get all cos 2cos 2 (x) + 2 cos(x)= 0 Combine Like Terms 2cos(x) [cos(x) +1]= 0 Factor So 2cos(x) = 0 and cos(x) +1 = 0 cos(x) = 0 cos(x) = -1 x = π 2 and 3π 2 x = π

EX 4 cont. Checking for extraneous solutions Because we squared the original equation, we must check for extraneous solutions

WS

NEXT DAY

Functions Involving Multiple Angles EX 1: Solve 2 cos(3x) -1 = 0 2 cos(3x) -1 = 0 Rewrote original equation 2 cos(3x) = 1 Added one to each side cos(3x) = 1 2 Divided each side by 2 When does cos(x) = 1 2 ? At π 3 and 5π 3 So you know that 3x = π 3 and 3x = 5π 3 general solutions would be 3x = π 3 + 2πn and 3x = 5π 3 + 2πn get x by itself x = π 9 + 2πn 3 and x = 5π 9 + 2πn 3

EX 2 Solve 3 tan 𝑥 2 + 3 = 0 3 tan( 𝑥 2 ) = -3 tan( 𝑥 2 ) = -1 So when does tan(x) = -1? On the interval [0, π), tan(x) = -1 at only 3π 4 +nπ So 𝑥 2 = 3π 4 +nπ multiply each side by 2 to get x = 3π 2 + 2nπ