Sum Difference Formulas

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Answer Key & Explanations

Sum Difference Formulas · Grade 11 · Worksheet 2

1. cos(71°)cos(26°) + sin(71°)sin(26°) = ? Answer: √2/2 Solution: Identify the formula: cos(A - B) = cos A cos B + sin A sin B. Here, A = 71°, B = 26°. The expression becomes cos(71° - 26°) = cos(45°).
   Full step-by-step solution

   Step 1: Identify the formula: cos(A - B) = cos A cos B + sin A sin B. Here, A = 71°, B = 26°. Step 2: The expression becomes cos(71° - 26°) = cos(45°). Step 3: cos(45°) = √2/2. The exact value is √2/2.

2. cos(112°) = cos(70° + 42°) = ? Answer: -cos(42°)cos(70°) + sin(42°)sin(70°) Solution: Step 1: Apply the cosine sum formula: cos(70° + 42°) = cos70°cos42° - sin70°sin42° Step 2: Determine the quadrant: 112° is in quadrant II where cosine is negative Step 3: Since 112° = 180° - 68°, we can also write cos(112°) = -cos(68°) Step 4: The exact expression using the sum formula is…
   Full step-by-step solution

   Step 1: Apply the cosine sum formula: cos(70° + 42°) = cos70°cos42° - sin70°sin42° Step 2: Determine the quadrant: 112° is in quadrant II where cosine is negative Step 3: Since 112° = 180° - 68°, we can also write cos(112°) = -cos(68°) Step 4: The exact expression using the sum formula is cos70°cos42° - sin70°sin42° Step 5: This can also be written as -cos(42°)cos(70°) + sin(42°)sin(70°) by rearranging terms The final answer is -cos(42°)cos(70°) + sin(42°)sin(70°)

3. sin(77°)cos(17°) - cos(77°)sin(17°) = ? Answer: √3/2 Solution: Identify the formula: sin(A - B) = sin A cos B - cos A sin B. Here, A = 77°, B = 17°. The expression simplifies to sin(77° - 17°) = sin(60°).
   Full step-by-step solution

   Step 1: Identify the formula: sin(A - B) = sin A cos B - cos A sin B. Here, A = 77°, B = 17°. Step 2: The expression simplifies to sin(77° - 17°) = sin(60°). Step 3: sin(60°) = √3/2. The exact value is √3/2.

4. A regular hexagon is inscribed in a circle with radius 10 cm. The hexagon is divided into 6 congruent equilateral triangles by drawing lines from the center to each vertex. Using trigonometric identities, find the exact area of one of these equilateral triangles. Answer: 25√3 Solution: In a regular hexagon inscribed in a circle, each side equals the radius. So side length s = 10 cm. Each triangle formed by connecting the center to adjacent vertices is equilateral with side length 10 cm.
   Full step-by-step solution

   Step 1: In a regular hexagon inscribed in a circle, each side equals the radius. So side length s = 10 cm. Step 2: Each triangle formed by connecting the center to adjacent vertices is equilateral with side length 10 cm. Step 3: The area of an equilateral triangle with side length s is (s²√3)/4. Step 4: Substitute s = 10: Area = (10²√3)/4 = (100√3)/4 = 25√3. The exact area of one equilateral triangle is 25√3 square centimeters.

5. Mere is a sound engineer analyzing the interference of two audio waves. The combined wave is modeled by the function f(t) = 12 sin(8t) + 16 cos(8t), where f(t) is the amplitude in decibels and t is time in seconds. To calibrate the system, she needs to rewrite this expression in the form R sin(8t + α) using the sine sum formula. What is the exact value of the amplitude R? Answer: 20 Solution: We want to write f(t) = 12 sin(8t) + 16 cos(8t) in the form R sin(8t + α). Using the sine addition formula: R sin(8t + α) = R sin(8t) cos α + R cos(8t) sin α.
   Full step-by-step solution

   Step 1: We want to write f(t) = 12 sin(8t) + 16 cos(8t) in the form R sin(8t + α). Step 2: Using the sine addition formula: R sin(8t + α) = R sin(8t) cos α + R cos(8t) sin α. Step 3: Matching coefficients with the given expression, we have: R cos α = 12 and R sin α = 16. Step 4: The amplitude R is found by squaring and adding these equations: (R cos α)^2 + (R sin α)^2 = 12^2 + 16^2. Step 5: This simplifies to R^2 (cos^2 α + sin^2 α) = 144 + 256. Step 6: Since cos^2 α + sin^2 α = 1, we have R^2 = 400. Step 7: Taking the positive square root (amplitude is positive): R = sqrt(400) = 20. Step 8: Therefore, the exact amplitude of the combined wave is 20 decibels. The answer is 20.

6. cos(50°)cos(20°) + sin(50°)sin(20°) = ? Answer: √3/2 Solution: Recognize this matches the cosine difference formula: cos(A - B) = cosA cosB + sinA sinB Identify A = 50° and B = 20° Apply the formula: cos(50° - 20°) = cos(30°) Evaluate cos(30°) = √3/2 The expression simplifies to √3/2
   Full step-by-step solution

   Step 1: Recognize this matches the cosine difference formula: cos(A - B) = cosA cosB + sinA sinB Step 2: Identify A = 50° and B = 20° Step 3: Apply the formula: cos(50° - 20°) = cos(30°) Step 4: Evaluate cos(30°) = √3/2 Step 5: The expression simplifies to √3/2

7. Olivia is analyzing a geometric pattern on a coordinate plane. A point P lies on the unit circle at an angle of 75 degrees from the positive x-axis. Using sum or difference formulas, determine the exact coordinates of point P. Answer: ( (sqrt(6) - sqrt(2))/4 , (sqrt(6) + sqrt(2))/4 ) Solution: The coordinates of point P are (cos 75°, sin 75°). Write 75° = 45° + 30°. Use the cosine sum formula: cos(A + B) = cos A cos B - sin A sin B.
   Full step-by-step solution

   Step 1: The coordinates of point P are (cos 75°, sin 75°). Step 2: Write 75° = 45° + 30°. Step 3: Use the cosine sum formula: cos(A + B) = cos A cos B - sin A sin B. cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2) = sqrt(6)/4 - sqrt(2)/4 = (sqrt(6) - sqrt(2))/4. Step 4: Use the sine sum formula: sin(A + B) = sin A cos B + cos A sin B. sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4. Step 5: Therefore, the exact coordinates of point P are ((sqrt(6) - sqrt(2))/4, (sqrt(6) + sqrt(2))/4).