Addition Formulas

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Answer Key & Explanations

Addition Formulas · Grade 11 · Worksheet 1

1. Using the angle addition formula for sine, find the exact value of sin(75°) by expressing it as sin(45° + 30°). Answer: (sqrt(6)+sqrt(2))/4 Solution: Step 1: Apply the sine addition formula: sin(45° + 30°) = sin45°cos30° + cos45°sin30° Step 2: Substitute known exact values: sin45° = sqrt(2)/2, cos30° = sqrt(3)/2, cos45° = sqrt(2)/2, sin30° = 1/2 Step 3: Calculate: (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6)/4) + (sqrt(2)/4) Step 4:…
   Full step-by-step solution

   Step 1: Apply the sine addition formula: sin(45° + 30°) = sin45°cos30° + cos45°sin30° Step 2: Substitute known exact values: sin45° = sqrt(2)/2, cos30° = sqrt(3)/2, cos45° = sqrt(2)/2, sin30° = 1/2 Step 3: Calculate: (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6)/4) + (sqrt(2)/4) Step 4: Combine terms: (sqrt(6) + sqrt(2))/4 The answer is (sqrt(6)+sqrt(2))/4.

2. sin(165°)cos(15°) - cos(165°)sin(15°) = ? Answer: 1/2 Solution: Recognize the trigonometric identity for sine of a difference: sin(A - B) = sin A cos B - cos A sin B Compare the given expression with the identity: sin(165°)cos(15°) - cos(165°)sin(15°) matches the right-hand side where A = 165° and B = 15° Apply the identity: sin(165° - 15°) = sin(150°)…
   Full step-by-step solution

   Step 1: Recognize the trigonometric identity for sine of a difference: sin(A - B) = sin A cos B - cos A sin B Step 2: Compare the given expression with the identity: sin(165°)cos(15°) - cos(165°)sin(15°) matches the right-hand side where A = 165° and B = 15° Step 3: Apply the identity: sin(165° - 15°) = sin(150°) Step 4: Simplify: 165° - 15° = 150° Step 5: Evaluate sin(150°) = 1/2 Step 6: The final answer is 1/2

3. Isabella is a marine biologist studying the path of a migrating whale pod. The pod's movement relative to a research station can be modeled by the vector sum of two displacement vectors: one 17 km at an angle of 82° north of east, and another 7 km at an angle of 37° north of east. To calculate the total displacement, Isabella needs to compute cos(82° - 37°) exactly. Using the cosine subtraction formula, prove that cos(82° - 37°) = cos(82°)cos(37°) + sin(82°)sin(37°), and then determine the exact value of cos(82° - 37°). Answer: √2/2 Solution: Recall the cosine subtraction formula: cos(A - B) = cos A cos B + sin A sin B. This proves the given identity. Simplify the angle: 82° - 37° = 45°.
   Full step-by-step solution

   Step 1: Recall the cosine subtraction formula: cos(A - B) = cos A cos B + sin A sin B. Step 2: Apply the formula with A = 82° and B = 37°: cos(82° - 37°) = cos(82°)cos(37°) + sin(82°)sin(37°). This proves the given identity. Step 3: Simplify the angle: 82° - 37° = 45°. Step 4: Therefore, cos(82° - 37°) = cos(45°). Step 5: The exact value of cos(45°) is √2/2. The answer is √2/2.

4. An architect is designing a triangular support structure for a bridge. She knows that one angle measures 75° and an adjacent angle measures 15°. Using the angle addition formula for sine, determine sin(75° + 15°) to verify the structural integrity calculation. What is the exact value of this trigonometric expression? Answer: 1 Solution: We are asked to find sin(75° + 15°) using the angle addition formula for sine. sin(A + B) = sin A cos B + cos A sin B Here, A = 75° and B = 15°.
   Full step-by-step solution

   Step 1: Understand the problem We are asked to find sin(75° + 15°) using the angle addition formula for sine. Step 2: Recall the angle addition formula for sine The formula is: sin(A + B) = sin A cos B + cos A sin B Step 3: Apply the formula Here, A = 75° and B = 15°. So: sin(75° + 15°) = sin 75° cos 15° + cos 75° sin 15° Step 4: Notice a simpler approach Actually, 75° + 15° = 90°, so we can compute directly: sin(75° + 15°) = sin(90°) Step 5: Evaluate sin(90°) From basic trigonometry, sin(90°) = 1. Step 6: Conclusion Therefore, sin(75° + 15°) = 1. Final answer: 1

5. cos(195°)cos(45°) + sin(195°)sin(45°) = ? Answer: -√3/2 Solution: Recognize the cosine subtraction identity: cos(A - B) = cos A cos B + sin A sin B Compare the given expression cos(195°)cos(45°) + sin(195°)sin(45°) with the identity.
   Full step-by-step solution

   Step 1: Recognize the cosine subtraction identity: cos(A - B) = cos A cos B + sin A sin B Step 2: Compare the given expression cos(195°)cos(45°) + sin(195°)sin(45°) with the identity. It matches the right-hand side where A = 195° and B = 45°. Step 3: Apply the identity: cos(195° - 45°) = cos(150°) Step 4: Simplify the angle: 195° - 45° = 150° Step 5: Evaluate cos(150°). Since 150° is in Quadrant II, cosine is negative. The reference angle is 180° - 150° = 30°, and cos(30°) = √3/2. Therefore, cos(150°) = -√3/2. Step 6: The final answer is -√3/2.

6. Aroha is a marine biologist studying wave patterns near a harbor. She models the height of a wave (in meters) over time using the function H(t) = 8 sin(πt/6 + π/3). To analyze the energy distribution, she needs to rewrite this function as a sum of sine and cosine functions. Using the angle addition formula for sine, express H(t) in the form A sin(πt/6) + B cos(πt/6), and determine the exact values of A and B. Answer: A = 4, B = 4√3 Solution: Use the angle addition formula: sin(X + Y) = sin X cos Y + cos X sin Y. In H(t) = 8 sin(πt/6 + π/3), let X = πt/6 and Y = π/3. Recall exact values: cos(π/3) = 1/2, sin(π/3) = √3/2.
   Full step-by-step solution

   Step 1: Use the angle addition formula: sin(X + Y) = sin X cos Y + cos X sin Y. Step 2: In H(t) = 8 sin(πt/6 + π/3), let X = πt/6 and Y = π/3. Step 3: Apply the formula: H(t) = 8 [ sin(πt/6) cos(π/3) + cos(πt/6) sin(π/3) ]. Step 4: Recall exact values: cos(π/3) = 1/2, sin(π/3) = √3/2. Step 5: Substitute: H(t) = 8 [ sin(πt/6)(1/2) + cos(πt/6)(√3/2) ]. Step 6: Distribute the 8: H(t) = 8*(1/2) sin(πt/6) + 8*(√3/2) cos(πt/6). Step 7: Simplify: H(t) = 4 sin(πt/6) + 4√3 cos(πt/6). Step 8: Therefore, A = 4 and B = 4√3. The answer is A = 4, B = 4√3.

7. A diagram on a coordinate plane shows a unit circle centered at the origin. Point P is at (1, 0). Point Q is obtained by rotating P counterclockwise by angle A = 71°, and point R is obtained by rotating Q counterclockwise by angle B = 26°. Using the coordinates of Q and R on the unit circle, prove the sine addition formula: sin(A + B) = sin A cos B + cos A sin B. Then, use this formula to find the exact value of sin(71° + 26°) expressed as a simplified sum of products of sine and cosine of 71° and 26°. Answer: sin 71° cos 26° + cos 71° sin 26° Solution: On the unit circle, point Q has coordinates (cos A, sin A) = (cos 71°, sin 71°). Point R is obtained by rotating Q by angle B, so R has coordinates (cos(A+B), sin(A+B)) = (cos 97°, sin 97°).
   Full step-by-step solution

   Step 1: On the unit circle, point Q has coordinates (cos A, sin A) = (cos 71°, sin 71°). Point R is obtained by rotating Q by angle B, so R has coordinates (cos(A+B), sin(A+B)) = (cos 97°, sin 97°). Step 2: Consider the rotation from Q to R. The coordinates of R can also be expressed by rotating the point (cos A, sin A) by angle B using the rotation matrix: (x', y') = (x cos B - y sin B, x sin B + y cos B). Step 3: Apply this rotation: x-coordinate of R = cos A cos B - sin A sin B = cos 71° cos 26° - sin 71° sin 26°. y-coordinate of R = sin A cos B + cos A sin B = sin 71° cos 26° + cos 71° sin 26°. Step 4: But we already know the y-coordinate of R is sin(A+B) = sin(71°+26°) = sin 97°. Therefore, sin(A+B) = sin A cos B + cos A sin B. Step 5: Substituting A = 71° and B = 26°: sin(71°+26°) = sin 71° cos 26° + cos 71° sin 26°. The answer is sin 71° cos 26° + cos 71° sin 26°.