(MATH-110) IB HIGHER
LEVEL STUDIES:
FUNCTIONS AND GRAPHS:
TRIGONOMETRY FINAL
EXAM STUDY GUIDE 2834
Study Ace Smart
, IB Higher Level Studies: Functions and Graphs (MATH-110)
Name: ___________________________
Instructor: ----
Date: August 24, 2025
Final Exam Study Guide
Solve the 20 problems below. Justify your answers and show all intermediate steps for full marks.
1. (5 points) Trigonometry: Polar to Cartesian Conversion
A point is given in polar coordinates (r, θ) as (4, 45°). Convert this point to Cartesian coordinates (x, y).
Solution:
Formulas: Recall the conversion formulas: x = r * cos(θ) and y = r * sin(θ).
Substitute: Plug in r and θ r = 4 and θ = 45°.
x = 4 * cos(45°) = 4 * 0.707 y
= 4 * sin(45°) = 4 * 0.707
Step 3:
Answer: The Cartesian coordinates are approximately (2.83, 2.83).
Pedagogical Insight: The sum-to-product and product-to-sum identities are less common but invaluable in signal processing
for analyzing wave interference.
2. (6 points) Right-Triangle Word Problem
A ladder of length 31 meters leans against a wall, making a 70° angle with the ground. How high up the wall
does the ladder reach? The diagram below (Figure 1) illustrates the setup.
Figure 2. Illustration for the problem.
Solution:
Step 1: Identify relationship. The building height (h) is opposite the angle, and the distance is adjacent. Use tan.
Equation: tan(70°) = h / 31
Calculation: h = 31 * sin(70°) = 29.13
Answer: The height is approximately 29.13 meters/feet.
Pedagogical Insight: Inverse trigonometric functions are critical for solving equations where the angle is the unknown
variable.
3. (4 points) Analytical Trigonometry: De Moivre's Theorem
A complex number in polar form is given by z = 4(cos(30°) + i*sin(30°)). The task is to compute the value of z^4
using De Moivre's Theorem.
Solution:
Recall Theorem: De Moivre's Theorem states:
[r(cos(θ) + i*sin(θ))]^n = r^n(cos(nθ) + i*sin(nθ)).
Step 2: Substitute values., θ=30, and n=4.
z^4 = 4^4(cos(4*30°) + i*sin(4*30°)).
Calculate:
z^4 = 256(cos(120°) + i*sin(120°)).
Answer: 256(cos(120°) + i*sin(120°))
Pedagogical Insight: Verifying trigonometric identities hones algebraic manipulation skills and deepens understanding of
the relationships between functions.
LEVEL STUDIES:
FUNCTIONS AND GRAPHS:
TRIGONOMETRY FINAL
EXAM STUDY GUIDE 2834
Study Ace Smart
, IB Higher Level Studies: Functions and Graphs (MATH-110)
Name: ___________________________
Instructor: ----
Date: August 24, 2025
Final Exam Study Guide
Solve the 20 problems below. Justify your answers and show all intermediate steps for full marks.
1. (5 points) Trigonometry: Polar to Cartesian Conversion
A point is given in polar coordinates (r, θ) as (4, 45°). Convert this point to Cartesian coordinates (x, y).
Solution:
Formulas: Recall the conversion formulas: x = r * cos(θ) and y = r * sin(θ).
Substitute: Plug in r and θ r = 4 and θ = 45°.
x = 4 * cos(45°) = 4 * 0.707 y
= 4 * sin(45°) = 4 * 0.707
Step 3:
Answer: The Cartesian coordinates are approximately (2.83, 2.83).
Pedagogical Insight: The sum-to-product and product-to-sum identities are less common but invaluable in signal processing
for analyzing wave interference.
2. (6 points) Right-Triangle Word Problem
A ladder of length 31 meters leans against a wall, making a 70° angle with the ground. How high up the wall
does the ladder reach? The diagram below (Figure 1) illustrates the setup.
Figure 2. Illustration for the problem.
Solution:
Step 1: Identify relationship. The building height (h) is opposite the angle, and the distance is adjacent. Use tan.
Equation: tan(70°) = h / 31
Calculation: h = 31 * sin(70°) = 29.13
Answer: The height is approximately 29.13 meters/feet.
Pedagogical Insight: Inverse trigonometric functions are critical for solving equations where the angle is the unknown
variable.
3. (4 points) Analytical Trigonometry: De Moivre's Theorem
A complex number in polar form is given by z = 4(cos(30°) + i*sin(30°)). The task is to compute the value of z^4
using De Moivre's Theorem.
Solution:
Recall Theorem: De Moivre's Theorem states:
[r(cos(θ) + i*sin(θ))]^n = r^n(cos(nθ) + i*sin(nθ)).
Step 2: Substitute values., θ=30, and n=4.
z^4 = 4^4(cos(4*30°) + i*sin(4*30°)).
Calculate:
z^4 = 256(cos(120°) + i*sin(120°)).
Answer: 256(cos(120°) + i*sin(120°))
Pedagogical Insight: Verifying trigonometric identities hones algebraic manipulation skills and deepens understanding of
the relationships between functions.
