# 7.3 Content - Solving Equations Using Trigonometric IDs

Quick Instructions
Add Solving Equations Using Trigonometric IDs content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.

Solving Equations Using Trignometric IDs (Unit 7B) Learning Targets

• Solve trigonometric equations using identities and inverse trig. relations
• Solve trigonometric equations involving multiples of angles

Evaluation is different from simplifying in the sense of accumulating an actual answer as opposed to verifying that the equations will be equal!

‍‍Ways to succeed in solving trig equations:
• Recognition
• A better background of new ways to solve trig equation
• Don't forget basic knowledge of trig identities
• DO NOT OVER COMPLICATE
• Getting frustrated doesnt help, sometimes there can be more ways then one to solve an equation
• Stay focused and you'll get through it!‍‍

Things To Remember When Solving Trig Equations:
• Be able to solve over an interval
• Don't forget the unit circle values
• Remember to isolate the x
• Don't forget to report all the solutions
• When the interval is over all real numbers be able to write in form of "All elements of the Real"

Solving Trig Equations Over a Given Domain:
Given Domain
• When given a domain, this will restrict the answer options
• A domain is given in instuction, otherwise there is none
• In strategies below, there are cases with restrictions and not to give further instruction

Strategies to show how to approach different types of trig equations:
Strategy 1
Are the known values in your equation on the unit circle?...
*Use triangles to help determine the values not on the unit circle*
Given
$\sin x = \frac{4}{5}$
and
$\sin y = \frac{8}{17}$
where 0° < x < 90° < y <180°. Evaluate sec(x+y).

It will help to understand this idea:
Names of expressions:
• 4/5 is the ratio
• ‍sin(x) is the angle which depicts the ratio given

Steps:
Step 1: Recognize sin(x) results in a 3-4-5 triangle in the first quadrant and sin(y) results in a 8-15-17 triangle in the second quadrant
Step 2: Recognize that secant is 1/cosine
Step 3: cos(x+y) changes to cos(x)cos(y) - sin(x)sin(y)
Step 4: Substitute the values you get for sin(x), cos(x), sin(y) and cos(y) which you find from the triangles

Given Restriction:
Restriction:
$\left [ -\pi ,\pi \right ]$
Meaning any answers given must be negative in quadrents 3 and four and positive in quadrents 1 and 2 (the numerical values have been given)
If you determined the angle, you would convert that angle to be in the restriction...
example of a solved for angle:
$\sin^{-1}(-4/5)$

$\sin^{-1}(-4/5)$
AND
$\pi -\sin^{-1}\left ( -4/5 \right )$
Strategy 2
*Rewrite as sum or difference of 2 known unit circle angle measures and then use sum/difference identities*
Solve: -cos(-285°)
=-cos(285°)
=-cos(240° + 45°)
=-[(cos240°)(cos45°) - (sin240°)(sin45°)]

$$=(\frac{1}{2})(\frac{\sqrt{2}}{2})-(\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) =\frac{\sqrt{2}-\sqrt{6}}4$ Steps: Step 1: Recognize the Even/Odd Identity of -cos(-x) = -cos(x) Step 2: Separate the given angle into 2 known unit circle angle measures Step 3: Apply the cos(x) sum identity of cos(A)cos(B) - sin(A)sin(B) Step 4: Substitute the unit circle values for the 2 angles Step 5: Evaluate the equation Strategy 3 *Sum/difference identities backwards* $\[cos(\frac{5\pi}{12})cos(\frac{\pi }{12})+sin(\frac{5\pi }{12})sin(\frac{\pi }{12})=cos(\frac{5\pi }{12}-\frac{\pi }{12})=cos(\frac{\pi }{3})=\frac{1}{2}$$

Step 1: Given an equation in the form cos(A)cos(B) + sin(A)sin(B), recognize that the equation is the equivalent of cos(A-B)
Step 2: Substitute the values of A and B into the equation cos(A-B)
Step 3: Evaluate the resulting expression, which is a unit circle value

Strategy 4
*Just solve with values on the unit circle*
Solve: sin(150°)cos(135°) - cos(150°)sin(135°).
$(\frac{1}{2})(\frac{-\sqrt{2}}{2})-(\frac{-\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) =-\frac{\sqrt{2}+6}{4}$

Steps:
Step 1: Recognize that the given angles are unit circle angle values, so cos(x) and sin(x) values for each angle are known
Step 2: Using the equation sin(A-B), plug in A and B values
Step 3: Evaluate the expression

Strategy 5
*Just solve with values on the unit circle*
*Sin(x)/cos(x) values must be on the unit circle, so sin(x) [-1,1] and cos(x) [-1,]*

$$sin(x)cos(2x)-cos(x)sin(2x)+3\csc (x)=2 over [0,4\pi ]\\sin(x-2x)+\frac{3}{sin(x)}-2=0\\-sin^2x-2sinx+3 =0 \\sin^2x+2sinx-3=0\\(sinx-1)(sinx+3)=0\\x=\frac{\pi }{2}, \frac{5\pi }{2}, -3 (extraneous)$$

Steps:
Step 1: Recognize sin(x)cos(2x) - cos(x)sin(2x) is the equivalent of sin(A-B) and that csc(x) is the inverse function of cos(x)
Step 2: Using even/odd identities, recognize that sin(x-2x) = sin(-x) = -sin(x)
Step 3: Multiply the entire equation by sin(x) to clear the fractions
Step 4: Recognize that the resulting equation is a factorable quadratic
Step 5: Factor the equation and set each binomial equal to 0
Step 6: Solve for each binomial
Step 7: Recognize that x=-3 is not a valid answer because sin(x) unit circle values are restricted to [-1,1]

Strategy 6
*Check answers with tan(x), cot(x), csc(x), or sec(x) in the original equation because if the denomintor = 0, then the solution is extraneous*
*Check the answers when you square an equation*
*Recognize interval given*

$$tan(x)sin(x)-cot(x)sin(x)+tan(x)-cot(x)=0 over [-\pi ,\pi ]\\(tanx-cotx)(sinx+1)=0\\tan(x)=cot(x) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; sin(x)=-1\\tan(x)=\frac{1}{tan(x)} \;\;\;\;\;\;\;\;\;\;\;\;\;\; x=Sin^{-1}(-1)\\tan^2x=1\\x=Tan^{-1}(\pm 1)\\x=\frac{-3\pi }{4}, \frac{-\pi}{2}(extraneous),\frac{-\pi}{4}, \frac{\pi}{4}, \frac{3\pi }{4}$$

Steps:
Step 1: Recognize that the given equation is factorable
Step 2: Upon factoring the equation, set each binomial equal to 0
Step 3: Use inverse trig to solve for x for each binomial
Step 4: Recognize that the second solution is extraneous because if that value is substituted in to the original equation, the equation becomes undefined due to the cot(x)

Strategy 7
*Recognize the square root gives 2 different values*
*Be able to use unit circle to find values of x*
*Recognize the interval given*

$\[8\cos^{2}x = 4 \;\; interval (0, 2\pi] \cos^{2}x = \frac{1}{2} \\ \cos x = \sqrt{\frac{1}{2}} \\ \cos x = \frac{\sqrt{2}}{2} \; and \; -\frac{\sqrt{2}}{2} \\ \\ x = \frac{\pi}{4} , \frac {3\pi}{4} , \frac {5\pi}{4} , \frac {7\pi}{4}$

Steps:
Step 1: Divide the equation by 8
Step 2: Square root both sides of the equation
Step 3: It forms two different to solve for the square root
Step 4: Find the values of the cosine on the unit circle
Step 5: Make sure the solutions are bounded within the interval

Strategy 8
*Recognize the two different binomials and be able to set them separately equal to 0*
*Use the inverse trig when the values are not located on the unit circle*
*Recognize the interval given*

Solve for all values of x over the given interval

$Interval \; (\infty, -\infty) \\ \\ (3\tan x + 2)(\tan x - 1) = 0 \\ \\ (3\tan x + 2) = 0 \\ \\ \tan x = \frac{-2}{3} \\ \\ x = \tan^{-1}(\frac{-2}{3}) + 2\pi \; and \; \tan^{-1}(\frac{-2}{3}) + \pi \\ \\ (\tan x - 1) = 0 \\ \\ Also \; \tan x = 1 \\ \\ x = \frac{\pi}{4} \; and \; \frac{5\pi}{4} \\ \;Given \; the \; values \; over \; (0, 2\pi] \; which \; can \; be \; used \; to \; determine \; the \; values \; over \; (\infty, -\infty) \\ x = \tan^{-1}(\frac{-2}{3}) + \pi n \; and \; \frac{\pi}{4} + \pi n \; over \; all \; elements \; of \; the \; real$

Steps:
Step 1: Set the two separate binomials to zero
Step 2: Solve for tan(x) for each binomial
Step 3: Based on those values, look for them on the unit circle
Step 4: For the first binomial, be able to recognize the use of inverse because it is not a value on the unit circle
Step 5: For the second binomial, the value is on the unit circle
Step 6: Need to find the value over the given interval
Step 7: Locate the point on the unit circle and look at how many times it needs to go around the circle to get to the next answer

Video Support:

Unit Circle Practice:
*If the unit circle values have gone out of your head, here is a great cheesy quizzing website to help out!*
Unit Circle Practice:

Other Sources:
*Other ways to learn parts of this information*
http://www.clarku.edu/~djoyce/trig/
http://www.math.com/tables/algebra/functions/trig/functions.htm
http://www.freemathhelp.com/trigonometry-help.html

‍‍There's an APPell for that:‍‍
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• Math Ref Free
• Trigonometry [pearson]
• MathPage

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~Tashi Hebel
Junior at Stevenson High School