Chapter 9 - Trig Applications & Proofs

Question 1

1When given an application, how do you write the equation of sinusoidal function? (8)

Answer 1

1. Identify whether to use sine/cosine: Stated
2. Find the Midline/Vertical Shift (D): Given or (Max + Min)/2
3. Find the Amplitude (A): (Max – Min) / 2, or (Max - Midline)
4. Find Period: If problem states “takes ___ to complete one cycle,” that value is the period (Period = 2π/B)
5. Find B‑Value (B): If period is known, solve for B (Period = 2π/B)
6. Identify Phase Shift (C)
7. If Needed to Convert to Cofunction: Phase shift of 1/4 of period
8. Plug A/B/C/D into: y = A sin(B(x – C)) + D or y = A cos(B(x – C)) + D

Question 2

When given an application, how do you graph sine/cosine? (7)

Answer 2

1. Start w/ given equation & window: y = A sin(B(x – C)) + D or y = A cos(B(x – C)) + D
2. Identify/Graph Midline: y = D (vertical shift)
3. Identify Period: 2π/B (given B in equation)
4. Label x‑axis & y‑axis according to given window (take half space)
5. On calculator, change MODE to RADIANS if equation is in terms of π
6. Adjust WINDOW settings: X Max = Period, Y Min = Midline - Amplitude, Y Max = Midline + Amplitude
7. Graph Function: Using graph/table to plot 5 key points per cycle

Question 3

You can use your equation, your graph, or your calculator to answer:
1. Points of intersection (4)
2. Periods of time above/below a value (4)
3. Maximum and minimum values (3)
4. When maximum/minimum occurs (4)
5. Values at specific times
6. When the function reaches a certain height

Answer 3

1. Point of Intersection (POI):
   - Enter first curve/line into Y1
   - Enter second curve/line into Y2
   - Use: 2nd → TRACE → Intersect.
   - X‑value of intersection is solution
2. Period of Time Above a Value
   - Enter trig function into Y1
   - Enter constant value into Y2
   - Find both intersection points
   - Time above value = x₂ – x₁
3. Maximum/Minimum Values
   - Maximum = Midline + Amplitude
   - Minimum = Midline – Amplitude
   - Or use graph: 2nd → TRACE → Maximum/Minimum
   - Additional Max/Min: Add Multiples of Period to Initial Min/Max
4. When Maximum/Minimum Occurs
   - Use drawn graph: 2nd → TRACE → Maximum/Minimum.
   - Cosine Equations: Max at start of cycle, min at C + (period/2)
   - Sine Equations: Max at x = C + (Period/4), Min at x = C + (3·Period/4)
5. Value at a Specific Time: Plug time into x of equation & solve for y
6. When the Function Reaches a Certain Height: Plug height into y & use inverse trig to solve for x

Question 4

What are somethings to keep in mind? (3)

Answer 4

1. When given a cosine/sine application, change mode to radians when equation is in terms of π, and degrees when equation is in normal numbers
2. If graphing more than one curve, label both lines
3. To Pull X - Value from Graph to Home screen: 2nd “-“
4. Remember to use H(t) if told to instead of Y

Question 5

How do you solve a Ferris Wheel word problem? (5)

Answer 5

1. Determine the Type of Function
   - Starts at Bottom: -Cos Function
   - Starts at Top: +Cos
   - Starts at Midline Going Up: +Sin
   - Starts at Midline Going Down: -Sin
2. Imagine Graph Depends on Distance Relative to Ground
3. Amplitude (A): Height of FW (Diameter + Height off Ground) - Midline, will be NEGATIVE (if starts at bottom)
4. B-Value (B)= 2π/period
5. Period = Time it takes to complete one full rotation
6. Midline (D) = (Diameter/2) + Height Off Ground

Question 6

What are and how do you solve trig proofs? (4/4/3)
(Definition & New Information, Tips, Past Information)

Answer 6

Def/New Info:
1. Problems where you must show left side = the right side
2. Use Reciprocal Identities: secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ
3. Use Quotient Identities: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
4. Use Pythagorean Identities: sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, cot²θ + 1 = csc²θ
Tips:
1. Try to rewrite EVERYTHING in terms of sine and cosine
2. Combine fractions on the same side of equation (via x, +, -)
3. Try to create a common denominator on both/opp sides of “=”
4. When given a squared trig function, use opportunity to factor in order to cancel terms in a fraction
5. If stuck, see if you can pull out a common trig function
Past Information:
1. Complex Fractions: Multiply entire expression by product of all denominators to eliminate fraction
2. Multiplying Fractions: Multiply num × num & denom × denom
3. Adding Fractions: Multiply each numerator by other denominator to combine