Final Assessment

Knowledge Assessment

Complete this formal assessment to evaluate your knowledge of trigonometric relationships, signs, periods, transformations, and asymptotes. Choose an answer for each question and submit at the bottom.

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Review your answers and step-by-step worked solutions below.
Question 1

Evaluating Primary Ratios

In a right-angled triangle, if sin(θ) =

513
and θ is an acute reference angle, what is the exact value of tan(θ)?

Worked Solution:
  • Identify side lengths from the sine ratio:
    sin(θ) =
    OppositeHypotenuse
    =
    513

    Let Opposite = 5 and Hypotenuse = 13.
  • Calculate the adjacent side using the Pythagorean identity:
    Adjacent = √(13² - 5²) = √(169 - 25) = √144 = 12
  • Formulate the tangent ratio:
    tan(θ) =
    OppositeAdjacent
    =
    512
  • Correct Answer: Option C
Question 2

Signs of Ratios Across Quadrants

If cos(θ) > 0 and csc(θ) < 0, in which of the four quadrants does the terminal arm of angle θ lie?

Worked Solution:
  • Deconstruct the given conditions to identify quadrant limits:
    • cos(θ) > 0 (positive cosine) occurs in Quadrants I and IV.
    • csc(θ) < 0 (negative cosecant/sine) occurs in Quadrants III and IV.
  • Find the intersection of these two sets:
    Only Quadrant IV satisfies both conditions simultaneously.
  • Correct Answer: Option D
Question 3

Calculating Wave Periodicity

Determine the exact period of the horizontal wave represented by the function:
y = tan(3θ)

Worked Solution:
  • Identify the horizontal frequency parameter from the function y = tan(bθ). Here, b = 3.
  • Recall the fundamental period of the parent tangent wave:
    Unlike sine and cosine (which repeat every 360°), the parent tangent function has a fundamental period of 180° (π radians).
  • Apply the period adjustment formula:
    Period =
    180°b
    =
    180°3
    = 60°
  • Correct Answer: Option B
Question 4

Vertical Graph Shifts

Determine the exact absolute minimum value (lowest y-coordinate) reached by the graph of:
y = -4 cos(θ) + 3

Worked Solution:
  • Identify transformational properties from the general equation y = a cos(θ) + d:
    • Midline vertical displacement (d) = 3.
    • Amplitude (|a|) = |-4| = 4.
  • A vertical reflection (negative sign) mirrors the cosine graph over its midline but does not affect the maximum and minimum boundaries.
  • Calculate the minimum boundary point:
    Minimum = Midline - Amplitude
    Minimum = 3 - 4 = -1
  • Correct Answer: Option C
Question 5

Identifying Graph Discontinuities

Which of the following functions exhibits a vertical asymptote precisely at the horizontal angle value θ = 180° (π)?

Worked Solution:
  • Analyze how vertical asymptotes occur in rational trigonometric functions:
    Asymptotes exist where the denominator of the quotient function resolves to zero.
  • Express key functions through primary operations:
    • tan(θ) =
    sin(θ)cos(θ)
    (Asymptotes where cos(θ) = 0)

    • csc(θ) =
    1sin(θ)
    (Asymptotes where sin(θ) = 0)

    • sec(θ) =
    1cos(θ)
    (Asymptotes where cos(θ) = 0)
  • Evaluate the denominators at the specified angle θ = 180°:
    • cos(180°) = -1 (Non-zero, so tangent and secant are defined).
    • sin(180°) = 0 (Zero, making cosecant undefined).
  • Thus, cosecant yields an asymptote at θ = 180°.
  • Correct Answer: Option B