Formulas - Evaluation 5

Question 1

Trigonometric Identities and proofs

Answer 1

remember (a+b)(a-b)= a²-b²
switch methods if you get an undefined value
if the value is in radians eg π/12 sub it for 3π/12 - 2π/12 = π/4 - π/6 to get values that exist on the unit circle

Question 2

Cosine sum and difference formulas

Answer 2

cos(a + b) = cos a cos b - sin a sin b
cos(a - b) = cos a cos b + sin a sin b

Question 3

Sine sum and difference formulas

Answer 3

sin(a + b) = sin a cos b + sin b cos a
sin(a - b) = sin a cos b - sin b cos a

Question 4

Tangent sum and difference formulas

Answer 4

tan (a + b) = [tan a tan b] / [ 1 - tan a tan b]

tan(a - b) = [tan a tan b] / [1 + tan a tan b]

Question 5

Double angle Formulas

Answer 5

sin(2θ) = 2 sinθcosθ
cos(2θ) = cos²θ - sin²θ
= 1 - 2sin²θ
= 2cos²θ - 1

Question 6

Product to sum Formulas

Answer 6

sin a sin b = 1/2 [cos(a-b) - cos(a+b)]
cos a cos b = 1/2 [cos(a-b) + cos(a+b)]
sin a cos b = 1/2 [sin(a+b) + sin(a-b)]

Question 7

Half angle Formulas

Answer 7

sin(a/2) = ±√(1-cosa)/2
cos(a/2) = ±√(1+cosa)/2

Question 8

θ Quadrant

Answer 8

ex
π/2< θ <π/4 hence in Q2
π/4< θ/2 < π/2 hence in Q1

Question 9

Inverse sine

Answer 9

D: -1 ≤ x ≤ 1
R: -π/2 ≤ y ≤ π/2

When solving eg. θ = sin⁻¹(1/2)
sinθ = 1/2
θ = π/6 or 30º
(angles found on unit circle)
if angle is not in range take equivalent angle from periodicity

Question 10

Inverse cosine

Answer 10

D: -1 ≤ x ≤ 1
R: 0 ≤ y ≤ π

When solving eg. θ = cos⁻¹(0)
cosθ = 0
θ = π/2 or 90º
(angles found on unit circle)
if angle is not in range take equivalent angle from periodicity

Question 11

Double inverse functions

Answer 11

cos⁻¹(cos(x)) = x or sin⁻¹(sin(x)) = x
given that x is in the correct domain of cos/sin (x), if not in the correct domain, find an equivalent angle to substitute and that angle becomes x

cos(cos⁻¹(x) = x or sin(sin⁻¹(x)) = x
given that x is in the correct range of cos/sin (x), if not in the correct domain answer is DNE

Question 12

Order of operations matter

Answer 12

2sinθcosθ means 2sinθ x cosθ
work out 2sinθ first then multiply it by cosθ

Question 13

Inverse functions out of range

Answer 13

If the θ value falls out of the given range then find the equivalent angle or value using the unit circle

Question 14

Trigonometric equation

Answer 14

if a range isn’t specified find all the correct angles in the unit circle then..

for sin - __ + 2Kπ
for cos - __ + 2Kπ
for tan - __ + Kπ

if a range is given test for all solutions in the given range

if range is specified in θ and the angle given is 2θ, then solve for 2θ first then divide by 2 to solve for θ, then test in given range and find all solutions

if sin/cos θ is greater than 1, θ for that solution DNE