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Addition Formulas

Grade 11 · Trigonometry · Worksheet 2

  1. Liam is designing a triangular support beam for a bridge. One of the angles in the triangle measures 105 degrees. To calculate the exact length of the opposite side using the law of sines, he needs the exact value of sin(105°). Using the angle addition formula for sine, express sin(105°) as a sum of two standard angles (from 30°, 45°, 60°, 90°) whose trigonometric values are known, and derive the exact simplified value of sin(105°). Answer: ______________
  2. Using the angle addition formula for sine, find the exact value of sin(75°) by expressing it as sin(45° + 30°). Answer: ______________
  3. cos(195°)cos(45°) + sin(195°)sin(45°) = ? Answer: ______________
  4. sin(115°)cos(25°) + cos(115°)sin(25°) = ? Answer: ______________
  5. A regular hexagon is inscribed in a circle of radius 10 cm. The hexagon is divided into 6 congruent equilateral triangles by drawing lines from the center to each vertex. Using trigonometric addition formulas, prove that the area of one of these triangles is exactly 25√3 cm². What is the total area of the hexagon?
    Answer: ______________
  6. sin(135°)cos(45°) + cos(135°)sin(45°) = ? Answer: ______________
  7. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter of length 2. If one acute angle measures 15°, use the angle addition formula to find the exact value of sin(15°) by expressing it as sin(45° - 30°). Answer: ______________
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Answer Key & Explanations

Addition Formulas · Grade 11 · Worksheet 2

  1. Liam is designing a triangular support beam for a bridge. One of the angles in the triangle measures 105 degrees. To calculate the exact length of the opposite side using the law of sines, he needs the exact value of sin(105°). Using the angle addition formula for sine, express sin(105°) as a sum of two standard angles (from 30°, 45°, 60°, 90°) whose trigonometric values are known, and derive the exact simplified value of sin(105°). Answer: (√6 + √2)/4 Solution: Express 105° as a sum of two standard angles: 105° = 60° + 45°. Recall exact values: sin(60°) = √3/2, cos(60°) = 1/2, sin(45°) = √2/2, cos(45°) = √2/2. Substitute: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2).
    Full step-by-step solution

    Step 1: Express 105° as a sum of two standard angles: 105° = 60° + 45°. Step 2: Apply the sine addition formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°). Step 3: Recall exact values: sin(60°) = √3/2, cos(60°) = 1/2, sin(45°) = √2/2, cos(45°) = √2/2. Step 4: Substitute: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2). Step 5: Multiply: sin(105°) = (√6)/4 + (√2)/4. Step 6: Combine fractions: sin(105°) = (√6 + √2)/4. The exact value of sin(105°) is (√6 + √2)/4.

  2. 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.

  3. 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.

  4. sin(115°)cos(25°) + cos(115°)sin(25°) = ? Answer: √3/2 Solution: Recognize that sin(115°)cos(25°) + cos(115°)sin(25°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Identify A = 115° and B = 25° Apply the formula: sin(115° + 25°) = sin(140°) Calculate sin(140°) = sin(180° - 40°) = sin(40°) sin(40°) = √3/2 The final answer is √3/2
    Full step-by-step solution

    Step 1: Recognize that sin(115°)cos(25°) + cos(115°)sin(25°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Step 2: Identify A = 115° and B = 25° Step 3: Apply the formula: sin(115° + 25°) = sin(140°) Step 4: Calculate sin(140°) = sin(180° - 40°) = sin(40°) Step 5: sin(40°) = √3/2 Step 6: The final answer is √3/2

  5. A regular hexagon is inscribed in a circle of radius 10 cm. The hexagon is divided into 6 congruent equilateral triangles by drawing lines from the center to each vertex. Using trigonometric addition formulas, prove that the area of one of these triangles is exactly 25√3 cm². What is the total area of the hexagon? Answer: 150√3 Solution: In a regular hexagon inscribed in a circle, each central angle is 360°/6 = 60°. Each triangle formed has two sides equal to the radius (10 cm) and an included angle of 60°.
    Full step-by-step solution

    Step 1: In a regular hexagon inscribed in a circle, each central angle is 360°/6 = 60°. Step 2: Each triangle formed has two sides equal to the radius (10 cm) and an included angle of 60°. Step 3: The area of one triangle can be found using the formula: Area = (1/2)ab sin(C), where a and b are the radii and C is the central angle. Step 4: Area of one triangle = (1/2) × 10 × 10 × sin(60°) = (1/2) × 100 × (√3/2) = 100√3/4 = 25√3 cm². Step 5: Since there are 6 congruent triangles, total hexagon area = 6 × 25√3 = 150√3 cm². The answer is 150√3.

  6. sin(135°)cos(45°) + cos(135°)sin(45°) = ? Answer: 0 Solution: Recognize that sin(135°)cos(45°) + cos(135°)sin(45°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Identify A = 135° and B = 45° Apply the formula: sin(135° + 45°) = sin(180°) Calculate sin(180°) = 0 The final answer is 0
    Full step-by-step solution

    Step 1: Recognize that sin(135°)cos(45°) + cos(135°)sin(45°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Step 2: Identify A = 135° and B = 45° Step 3: Apply the formula: sin(135° + 45°) = sin(180°) Step 4: Calculate sin(180°) = 0 Step 5: The final answer is 0

  7. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter of length 2. If one acute angle measures 15°, use the angle addition formula to find the exact value of sin(15°) by expressing it as sin(45° - 30°). Answer: (√6 - √2)/4 Solution: We are given: sin(15°) = sin(45° - 30°). sin(A - B) = sin A cos B - cos A sin B. Let A = 45°, B = 30°.
    Full step-by-step solution

    We are given: sin(15°) = sin(45° - 30°). We use the sine subtraction formula: sin(A - B) = sin A cos B - cos A sin B. Let A = 45°, B = 30°. Step 1: Write the formula with these angles. sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°. Step 2: Recall exact values. sin 45° = √2 / 2 cos 45° = √2 / 2 sin 30° = 1/2 cos 30° = √3 / 2 Step 3: Substitute into the formula. sin(45° - 30°) = (√2 / 2) * (√3 / 2) - (√2 / 2) * (1/2) Step 4: Simplify each term. First term: (√2 / 2) * (√3 / 2) = (√2 * √3) / (2 * 2) = √6 / 4 Second term: (√2 / 2) * (1/2) = √2 / 4 So we have: sin(15°) = √6 / 4 - √2 / 4 Step 5: Combine into a single fraction. sin(15°) = (√6 - √2) / 4 This is the exact value of sin(15°). Final answer: (√6 - √2)/4