Download Trig Eq-Id Extra Review

Math 30-1
Trig Equations & Identities
Trigonometric Equations & Identities
Extra Review Session
Trig Equations
The solution to a trig equation is an angle, either in degrees or radians. This means that, after
factoring or isolating the trig ratio, you still need to find the angle that makes the simplified
equation true, by:
 special triangles, if possible
 unit circle (for quadrantal angles)
 using the sin-1, cos-1, tan-1 buttons on your calculator
 Don’t forget to use the CAST Rule!
Most trig equations have two solutions between 0 and 2 (0° and 360°).
 Remember to pay attention to the domain – if you need solution(s) outside of the “usual”
domain, find the appropriate coterminal angle(s) by adding or subtracting 2 or 360°.
 For a general solution, add multiples of the period (with n) to the solution(s) in the first
period.
Class Example:
Solve the equation 3 csc tan   5 tan   0
a) for the domain  2    
b) general solution in radians
Math 30-1
Trig Equations & Identities
Try These:
1. Solve the equation 2 cot 2   2 cot   3  cot 2  for the domain  360    0 .
2. Give the general solution to the equation 4 sin 2 x  3  0 in radians. (Try to write the solution
in the simplest possible form – it is possible to write it with a single statement!)
Some trig equations require using a trig identity to simplify it first. Use this as a last resort in the
following cases ONLY:
 It contains more than one trig ratio and cannot be factored
 It contains a double angle
Math 30-1
Trig Equations & Identities
Class Example:
Find the general solution to the equation 5 sin x  cos 2x  3  0 in degrees.
Try These:
3. Solve the following equations for the domain 0    2 :
a) 2 cos  7  3sec
b)
2 sin 2x  2 sin x  0
For more practice solving trig equations:
p. 211-214 #3, 5, 7, 9, 11, 13, 16, 22
p. 320-321 #1-6, 8-9, 11, 14, 16, (19), C2
Math 30-1
Trig Equations & Identities
Trig Identities
Here’s a quick rundown on the different things you may be asked to do with identities:
 Verify
- Plug in the value given on each side of the identity.
- Use exact values (special triangles) and the CAST Rule.
- DON’T use identities.
- The two sides should equal the same thing in the end. The answers will be trig ratios:
expressions with fractions and square roots, not  . You may need to rationalize the
denominator to get both sides to equal each other.
 Find NPVs (restrictions)
- Denominators  0 .
- This includes “hidden” denominators of tan, csc, sec, cot.
- Solve these just like trig equations with a general solution.
 Simplify/Prove
- Substitute in identities.
- Look for:
1. Squared terms – use Pythagorean identities (or possibly cos double angle)
2. Double angles – use double-angle formulas
3. Sum/difference identities
4. tan, csc, sec, cot – change to sin and cos
- Techniques:
1. Add/subtract fractions – need common denominator
2. Divide fractions – multiply by reciprocal
3. Factor
4. Multiply numerator and denominator by the conjugate
- When proving, remember to work on each side separately.
Class Example:
Consider the identity cot x 
a) Find the NPVs.
cos 2 x  1
.
sin 2 x
Math 30-1
Trig Equations & Identities
b) Verify that cot x 
cos 2 x  1
5
for x 
.
sin 2 x
6
c) Prove the identity algebraically.
Math 30-1
Trig Equations & Identities
Try These:
4. The expression
A.
B.
C.
D.
cot x  csc x
is equivalent to
sec x  1
sin x
tan x
csc x
cot x
5. For the following identities, write the NPVs, verify for   60 , and prove algebraically:
cot 
1
a)
csc  cos 
Math 30-1
Trig Equations & Identities
b)
cot 
1
 csc 2 
sin 2 2
c)
sin  cos  1  cos 

(This one is difficult!)
1  cos 
tan 
For more practice on trig identities:
p. 322-323 #1-4, 11-15
p. 314-315 #1-8, 11-13
Math 30-1
Trig Equations & Identities
There are a few more types of questions to remember with sum, difference, and double angle
identities:
 Write as a single trig function
- It’s usually either a sum/difference identity or a double angle identity – use your
formula sheet to go from the complicated version to the simple version
Class Example:
Write 1  2 sin 2
3
as a single trig function, then find the exact value.
8
Try These:
6. Write the following as a single trig ratio:
a) sin 179 cos 43  cos179 sin 43

3
5
5
b)

3
1  tan tan
5
5
tan

 tan
Find exact value
- Two types:
1. Given a trig ratio
- Use sum/difference identities to break it apart into the complex form – usually
sin or cos
- You will have to use x 2  y 2  r 2 to find the exact value of another trig ratio
- Plug in the value of each ratio and simplify. Don’t forget the CAST Rule.

2. Given an angle with a reference angle of
or 15°
12
- First, check to see if it’s a double angle identity. (These are the easy ones!)
- If it’s not a double angle identity, find two angles that add or subtract to the given
ratio
- Use sum/difference identities to break it apart into its complex form
- Then use special triangles and the CAST Rule to find the exact value of each ratio
and simplify the answer.
Math 30-1
Trig Equations & Identities
Class Example:
Find the exact value of the following:
5
a) cos A  45 , where cos A  and 270  A  360
8
b) sin
5
12
Try These:
7. Find the exact value of tan 195 .
Math 30-1
Trig Equations & Identities
4
 3

   , where  is in quadrant III and tan   .
8. Find the exact value of sin 
3
 4

For more practice on sum, difference, and double angle identities:
p. 306-307 #1-2, 4, 7-8, 11, 19-20
Answers:
1.   18,  135,  198,  315

2. x    n , n  I
3
 5
3
5
, ,
3. a)   ,
b) x  0,
3 3
4
4
4. D
5. a)   90n, n  I ; both sides equal 1
2
b)   90n, n  I ; both sides equal
3
c)   90n, n  I ; both sides equal
6. a) sin 136
7. tan 195 
b) tan
3 1
1 3
4
5
or 2  3
7 2
7
 3

   
8. sin 
or 
10
5 2
 4

3
; Hint for proof: multiply by the conjugate
6