Sum Difference Formulas

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Sum Difference Formulas · Grade 11 · Worksheet 1

1. Mason is analyzing a geometric pattern on a coordinate plane. A right triangle is inscribed in a unit circle with its hypotenuse as the diameter. One acute angle of the triangle measures 75°. Using a sum or difference formula, determine the exact coordinates of the vertex opposite the hypotenuse (the point on the circle that is not an endpoint of the diameter). Answer: (cos 75°, sin 75°) = ((√6 - √2)/4, (√6 + √2)/4) Solution: Recognize that the vertex opposite the hypotenuse is at angle 75° from the positive x-axis on the unit circle. Its coordinates are (cos 75°, sin 75°). Express 75° as 45° + 30°.
   Full step-by-step solution

   Step 1: Recognize that the vertex opposite the hypotenuse is at angle 75° from the positive x-axis on the unit circle. Its coordinates are (cos 75°, sin 75°). Step 2: Express 75° as 45° + 30°. Step 3: Use the sum formulas: cos(A + B) = cos A cos B - sin A sin B sin(A + B) = sin A cos B + cos A sin B Step 4: Apply with A = 45°, B = 30°: cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4) - (√2/4) = (√6 - √2)/4 sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4 Step 5: Therefore, the exact coordinates are ((√6 - √2)/4, (√6 + √2)/4).

2. Mason is a structural engineer analyzing a triangular truss for a bridge design. The truss forms an angle of 105° at one of its joints. Using the sum and difference formulas, determine the exact value of cos(105°). Answer: (√2 - √6)/4 Solution: Express 105° as a sum of two common angles: 105° = 60° + 45° Apply the cosine sum formula: cos(A + B) = cosA cosB - sinA sinB Here, A = 60° and B = 45° cos(60°) = 1/2 cos(45°) = √2/2 sin(60°) = √3/2 sin(45°) = √2/2 cos(105°) = cos(60° + 45°) = cos(60°)cos(45°) - sin(60°)sin(45°) = (1/2)(√2/2) -…
   Full step-by-step solution

   Step 1: Express 105° as a sum of two common angles: 105° = 60° + 45° Step 2: Apply the cosine sum formula: cos(A + B) = cosA cosB - sinA sinB Here, A = 60° and B = 45° Step 3: Substitute the exact values: cos(60°) = 1/2 cos(45°) = √2/2 sin(60°) = √3/2 sin(45°) = √2/2 Step 4: Apply the formula: cos(105°) = cos(60° + 45°) = cos(60°)cos(45°) - sin(60°)sin(45°) = (1/2)(√2/2) - (√3/2)(√2/2) = √2/4 - √6/4 Step 5: Combine terms: cos(105°) = (√2 - √6)/4 The answer is (√2 - √6)/4.

3. Emma is studying a geometric diagram on a coordinate plane. A point P lies on the unit circle such that the ray from the origin to P makes an angle of 75° with the positive x-axis. Using the sum formula for sine or cosine, determine the exact coordinates of point P. Answer: ( (sqrt(6)-sqrt(2))/4 , (sqrt(6)+sqrt(2))/4 ) Solution: On the unit circle, point P has coordinates (cos 75°, sin 75°). Write 75° = 45° + 30°.
   Full step-by-step solution

   Step 1: On the unit circle, point P has coordinates (cos 75°, sin 75°). Step 2: Write 75° = 45° + 30°. Step 3: Use the sum formulas: cos(A+B) = cos A cos B - sin A sin B sin(A+B) = sin A cos B + cos A sin B Step 4: For cos 75°: cos 75° = cos(45°+30°) = cos45°cos30° - sin45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = √6/4 - √2/4 = (√6 - √2)/4 Step 5: For sin 75°: sin 75° = sin(45°+30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4 Step 6: Therefore, the exact coordinates of P are ((√6-√2)/4, (√6+√2)/4).

4. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter. If one acute angle measures 30°, what are the exact coordinates of the vertex opposite the hypotenuse? Answer: (√3/2, 1/2) Solution: When a right triangle is inscribed in a circle with its hypotenuse as the diameter, the vertex opposite the hypotenuse lies on the circumference.
   Full step-by-step solution

   When a right triangle is inscribed in a circle with its hypotenuse as the diameter, the vertex opposite the hypotenuse lies on the circumference. On the unit circle, coordinates can be expressed using cosine and sine of the angle formed with the positive x-axis. This geometric property connects triangle geometry with circular functions.

5. Noah is an architect designing a curved glass canopy for a modern building. The canopy's cross-section is modeled by the function h(x) = 11 sin(x) + 6 cos(x), where h is the height in meters and x is the horizontal distance in radians from the starting point. To ensure the structure meets clearance requirements, Noah needs to find the exact maximum height of the canopy. Using the sine sum formula, rewrite h(x) in the form R sin(x + α) and determine the exact maximum height. Answer: √157 Solution: We want to write h(x) = 11 sin x + 6 cos x in the form R sin(x + α). Using the sine addition formula: R sin(x + α) = R sin x cos α + R cos x sin α. Matching coefficients, we have: R cos α = 11 and R sin α = 6.
   Full step-by-step solution

   Step 1: We want to write h(x) = 11 sin x + 6 cos x in the form R sin(x + α). Step 2: Using the sine addition formula: R sin(x + α) = R sin x cos α + R cos x sin α. Step 3: Matching coefficients, we have: R cos α = 11 and R sin α = 6. Step 4: The amplitude R is found by squaring and adding: (R cos α)^2 + (R sin α)^2 = 11^2 + 6^2 = 121 + 36 = 157. Step 5: Since (R cos α)^2 + (R sin α)^2 = R^2(cos^2 α + sin^2 α) = R^2, we get R^2 = 157. Step 6: Therefore R = √157 (positive amplitude). The exact maximum height of the canopy is √157 meters.

6. Emma is analyzing sound wave interference patterns in her physics lab. She has two sound waves represented by the functions f(t) = 3sin(2t) and g(t) = 4cos(2t), where t is time in seconds. Using trigonometric identities, determine the amplitude of the combined wave h(t) = f(t) + g(t) when expressed in the form Rsin(2t + α). Answer: 5 Solution: Write the combined wave function: h(t) = 3sin(2t) + 4cos(2t) We want to express this in the form Rsin(2t + α) = R[sin(2t)cos(α) + cos(2t)sin(α)] Compare coefficients: Rcos(α) = 3 and Rsin(α) = 4 Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 3² + 4² R²(cos²(α) + sin²(α)) = 9 + 16…
   Full step-by-step solution

   Step 1: Write the combined wave function: h(t) = 3sin(2t) + 4cos(2t) Step 2: We want to express this in the form Rsin(2t + α) = R[sin(2t)cos(α) + cos(2t)sin(α)] Step 3: Compare coefficients: Rcos(α) = 3 and Rsin(α) = 4 Step 4: Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 3² + 4² Step 5: R²(cos²(α) + sin²(α)) = 9 + 16 Step 6: Since cos²(α) + sin²(α) = 1, we get R² = 25 Step 7: Therefore, R = 5 Step 8: The amplitude of the combined wave is 5 Step 9: The answer is 5

7. Isabella is examining a geometric diagram on a coordinate plane. A regular pentagon is inscribed in a unit circle centered at the origin, with one vertex at (1, 0). The vertices are labeled A (at (1,0)), B, C, D, and E in counterclockwise order. Using sum or difference formulas for sine or cosine, determine the exact y-coordinate of vertex C. Answer: sin 144° = sin(180° - 36°) = sin 36° = (√(10 - 2√5))/4 Solution: A regular pentagon inscribed in a unit circle has 5 equal central angles. The central angle between consecutive vertices is 360°/5 = 72°. Vertex A is at angle 0° (coordinate (1,0)).
   Full step-by-step solution

   Step 1: A regular pentagon inscribed in a unit circle has 5 equal central angles. The central angle between consecutive vertices is 360°/5 = 72°. Step 2: Vertex A is at angle 0° (coordinate (1,0)). Vertex B is at 72°, vertex C is at 144°, vertex D is at 216°, vertex E is at 288°. Step 3: The y-coordinate of vertex C is sin 144°. Step 4: Use the difference formula: 144° = 180° - 36°. sin(180° - θ) = sin θ. So sin 144° = sin 36°. Step 5: The exact value of sin 36° is known: sin 36° = √(10 - 2√5) / 4. Step 6: Therefore, the exact y-coordinate of vertex C is √(10 - 2√5) / 4.