Chapter 9 - Sum & Difference, Double Angles, Inverse Trig

Question 1

What is Sum and Difference of Angles? (3/3/3)
(When to use it, Sum Formulas, Difference Formulas)

Answer 1

• Used when asked for exact value of a trigonometric function whose REFERENCE angle cannot be found on the unit circle or table
• Break up angle into sum (or difference) of reference angles whose exact value is on unit circle/table (0, 30, 45, 60, 90, 180, 270, 360)
• ## Rules can be applied to proofs as well when given a sum/difference in angles of trigonometric functionsSum:
  a) sin (A + B) = sin A ⋅ cos B + cos A ⋅ sin B
  b) cos (A + B) = cos A ⋅ cos B - sin A ⋅ sin B
  c) tan (A + B) = (tan A + tan B)/(1 - tan A ⋅ tan B)
  ——————————————————————————————–
  Difference (Just Flip the Signs):
  a) sin (A - B) = sin A ⋅ cos B - cos A ⋅ sin B
  b) cos (A - B) = cos A ⋅ cos B + sin A ⋅ sin B
  c) tan (A - B) = (tan A - tan B)/(1 + tan A ⋅ tan B)

Question 2

What are Double Angles? (3/5)
(Used When, Formulas)

Answer 2

• Used when given a trigonometric function and “2A”
• Use to find exact value when given exact value of just trig function of A
• ## Use when found in proofs/equations to put all functions in the same termsFormulas:
  1. Sin2A = 2 ⋅ sinA ⋅ cosA
  2. tan2A = (2tanA)/(1 - tan²A)
  3. Cos2A (Memorize first one and replace cos²A/sin²A w/ trig identities to find/use the other two equations):
  –> cos2A = cos²A - sin²A
  –> cos2A = 2cos²A - 1
  –> cos2A = 1 - 2sin²A

Question 3

What is the parent function of inverse sine? (4)

Answer 3

1. Domain: [-1, 1]
2. Range: [-π/2, π/2]
3. Points: (-1, -π/2), (-0.5, -π/6), (0, 0), (0.5, π/6), (1, π/2)
4. Reminders: Increases Left to Right, Do NOT use Arrows

Question 4

What is the parent function of inverse cosine? (4)

Answer 4

1. Domain: [-1, 1]
2. Range: [0, π]
3. Points: (-1, π), (-0.5, 2π/3), (0, π/2), (0.5, π/3), (1, 0)
4. Reminders: Decreases Left to Right, Do NOT use Arrows

Question 5

What is the parent function of inverse tangent? (4)

Answer 5

1. Domain: ALL REALS
2. Range: (-π/2, π/2)
3. Points: HA, (-1, -π/4), (0, 0), (1, π/4), HA
4. Reminders: Increases Left to Right, Include Arrows, Two Horizontal Asymptotes at the Top/Bottom

Question 6

What are general rules when evaluating inverse trigonometric functions? (3/3/1)

Answer 6

1. Domain/Range of (sin⁻¹x, csc⁻¹x, tan⁻¹x):
   
   • Angles are in [-π/2, π/2]
   • Quadrants: I and IV
   • Answers may be in degrees or radians (follow the problem)
2. Domain/Range of (cos⁻¹x, cot⁻¹x, sec⁻¹x):
   
   • Angles are in [0, π]
   • Quadrants: I and II
   • Answers may be in degrees or radians (follow the problem)
3. When given an inverse trig expression:
   
   • Convert angle → exact value OR exact value → angle

Question 7

How do you evaluate inverse trigonometric functions? (2/3/3)

Answer 7

When Exact Value is in Parentheses of Inverse Trig:
1. Convert inside value into an angle:
- Check if the value matches a special angle on the unit circle
- If NOT:
- Determine correct quadrant based on inverse trig function’s range
- Draw right triangle using given ratio
- Use triangle to evaluate outer trig function
Ex: sin(arctan(-1/4))
- Let θ = arctan(-1/4) → tanθ = -1/4 (not a special angle)
- arctan outputs angles in Q1 or Q4 → negative → Q4
- Triangle: O = -1, A = 4, FIND: hypotenuse = √17
- sinθ = opposite/hypotenuse = -1/√17
2. When value inside parentheses is negative:
- Angle must be in allowed quadrant where trig ratio is neg
- (cos, cot, sec → negative in Quadrant II)
- (sin, csc, tan → negative in Quadrant IV → negative angle)
3. After inverse trig, outer function:
- Convert Radians to Degrees inside ( )
- Find Reference Angle & ASTC (whether positive or negative)
- Find Exact Value
Ex:
- csc [sin⁻¹(-1/2)]
- csc [-π/6]
(convert radians to degrees to make it easier)
- csc [330]
- reference angle: 30 degrees, ASTC: Negative in Q4
- Exact Value = -1/2

Question 8

What are cofunctions?

Answer 8

1. (Sine & Cosine), (Cosecant, Secant), (Tangent, Cotangent)
2. Cofunctions whose Angles are Complementary have the same Exact Value