(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)

(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. 4. (4 points) Trigonometry: Polar to Cartesian Conversion A point is given in polar coordinates (r, θ) as (5, 135°). Convert this point to Cartesian coordinates (x, y). Solution: Formulas: The formulas are: x = r * cos(θ) and y = r * sin(θ). Step 2: Plug in r and θ r = 5 and θ = 135°. x = 5 * cos(135°) = 5 * -0.707 y = 5 * sin(135°) = 5 * 0.707 Calculate: Answer: The Cartesian coordinates are approximately (-3.54, 3.54). Pedagogical Insight: Understanding the unit circle is the absolute foundation for all of analytic trigonometry. 5. (9 points) Trigonometry: Verify Identity Verify the following trigonometric identity: (sec(x) - cos(x)) / sec(x) = sin²(x) Solution: LHS: We will manipulate the LHS. and express in terms of sin and cos. sec(x) = 1/cos(x). LHS = ( (1/cos(x)) - cos(x) ) / (1/cos(x)) Step 2: Combine terms in the numerator. LHS = ( (1 - cos²(x)) / cos(x) ) / (1/cos(x)) Step 3: Multiply by the reciprocal. LHS = ( (1 - cos²(x)) / cos(x) ) * (cos(x) / 1) Step 4: LHS = 1 - cos²(x) Step 5: Apply the Pythagorean identity. 1 - cos²(x) = sin²(x). This equals the right-hand side (RHS). Answer: The identity is verified as LHS = RHS. Pedagogical Insight: The Law of Sines is a powerful tool for solving oblique triangles, commonly used in surveying and navigation.