Sum Difference Formulas

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Sum Difference Formulas · Grade 11 · Worksheet 3

1. A right triangle is inscribed in a circle with radius 5 units. The triangle's hypotenuse lies along the diameter of the circle, and one of its acute angles measures 30°. What is the exact length of the side opposite the 30° angle? Answer: 5 Solution: A right triangle is inscribed in a circle of radius 5. The hypotenuse is the diameter of the circle. Identify the given angle.
   Full step-by-step solution

   Step 1: Understand the problem setup. A right triangle is inscribed in a circle of radius 5. The hypotenuse is the diameter of the circle. So, the hypotenuse length = 2 × radius = 2 × 5 = 10. Step 2: Identify the given angle. One acute angle is 30°. In a right triangle, the side opposite the 30° angle is the shortest side. Step 3: Recall the 30-60-90 triangle side ratios. In a 30-60-90 triangle: - Side opposite 30° = (1/2) × hypotenuse - Side opposite 60° = (√3/2) × hypotenuse - Hypotenuse = twice the shortest side. Step 4: Apply the ratio. Hypotenuse = 10. Side opposite 30° = (1/2) × hypotenuse = (1/2) × 10 = 5. Step 5: Final answer. The length of the side opposite the 30° angle is exactly 5 units.

2. Matiu is analyzing a geometric pattern formed by two intersecting circles. The circles have equal radii of 14 units and their centers are 14√3 units apart. Consider triangle formed by the two centers and one intersection point of the circles. Using sum or difference formulas for sine or cosine, find the exact value of cos(∠A) where ∠A is the angle at one of the centers in this triangle. Answer: √3/2 Solution: Draw the triangle. Let O1 and O2 be the centers, each 14 units from the intersection point P. O1O2 = 14√3 units.
   Full step-by-step solution

   Step 1: Draw the triangle. Let O1 and O2 be the centers, each 14 units from the intersection point P. O1O2 = 14√3 units. The triangle O1O2P is isosceles with sides O1P = O2P = 14 and base O1O2 = 14√3. Step 2: Use the Law of Cosines at vertex O1 (angle ∠O2O1P = θ). The side opposite θ is O2P = 14. The sides adjacent to θ are O1O2 = 14√3 and O1P = 14. Law of Cosines: (O2P)^2 = (O1O2)^2 + (O1P)^2 - 2(O1O2)(O1P) cos θ Step 3: Substitute the known lengths: 14^2 = (14√3)^2 + 14^2 - 2(14√3)(14) cos θ Step 4: Simplify each term: 196 = (196 * 3) + 196 - 2(196√3) cos θ 196 = 588 + 196 - 392√3 cos θ 196 = 784 - 392√3 cos θ Step 5: Solve for cos θ: 392√3 cos θ = 784 - 196 392√3 cos θ = 588 cos θ = 588 / (392√3) cos θ = (588 ÷ 196) / (392√3 ÷ 196) cos θ = 3 / (2√3) Rationalize: cos θ = (3√3) / (2 * 3) = √3 / 2 Step 6: Recognize that √3/2 = cos 30°. Therefore θ = 30°. Step 7: The problem asks for cos(∠A) where ∠A = θ. So cos(∠A) = √3/2. The exact value is √3/2.

3. sin(127°) = sin(90° + 37°) = ? Answer: cos(37°) Solution: Step 1: Apply the sum formula for sine: sin(90° + 37°) = sin(90°)cos(37°) + cos(90°)sin(37°) Step 2: Evaluate known trigonometric values: sin(90°) = 1 and cos(90°) = 0 Step 3: Substitute these values: (1)(cos(37°)) + (0)(sin(37°)) = cos(37°) + 0 Step 4: Simplify: cos(37°) The exact value is…
   Full step-by-step solution

   Step 1: Apply the sum formula for sine: sin(90° + 37°) = sin(90°)cos(37°) + cos(90°)sin(37°) Step 2: Evaluate known trigonometric values: sin(90°) = 1 and cos(90°) = 0 Step 3: Substitute these values: (1)(cos(37°)) + (0)(sin(37°)) = cos(37°) + 0 Step 4: Simplify: cos(37°) The exact value is cos(37°).

4. cos(285°) = ? Answer: (√6 - √2)/4 Solution: Express 285° as a sum of two special angles: 285° = 225° + 60°. Find exact values: cos(225°) = -√2/2, sin(225°) = -√2/2, cos(60°) = 1/2, sin(60°) = √3/2.
   Full step-by-step solution

   Step 1: Express 285° as a sum of two special angles: 285° = 225° + 60°. Step 2: Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB, with A = 225°, B = 60°. Step 3: Find exact values: cos(225°) = -√2/2, sin(225°) = -√2/2, cos(60°) = 1/2, sin(60°) = √3/2. Step 4: Substitute: cos(285°) = (-√2/2)(1/2) - (-√2/2)(√3/2) = -√2/4 + (√2·√3)/4 = -√2/4 + √6/4 = (√6 - √2)/4. The exact value of cos(285°) is (√6 - √2)/4.

5. Sophia is working on a design for a new roller coaster. The path of one section of the track is modeled by the function h(x) = 8sin(x) + 6cos(x), where h is the height in meters and x is the horizontal distance from the start in radians. To determine the maximum height of this section, Sophia needs to rewrite this function in the form Rsin(x + alpha). Using sum and difference formulas, what is the exact value of the amplitude R? Answer: 10 Solution: We want to write 8 sin(x) + 6 cos(x) in the form R sin(x + alpha). The sum formula for sine is: R sin(x + alpha) = R [sin(x)cos(alpha) + cos(x)sin(alpha)] = (R cos(alpha)) sin(x) + (R sin(alpha)) cos(x).
   Full step-by-step solution

   Step 1: We want to write 8 sin(x) + 6 cos(x) in the form R sin(x + alpha). Step 2: The sum formula for sine is: R sin(x + alpha) = R [sin(x)cos(alpha) + cos(x)sin(alpha)] = (R cos(alpha)) sin(x) + (R sin(alpha)) cos(x). Step 3: Equate coefficients with 8 sin(x) + 6 cos(x): R cos(alpha) = 8 R sin(alpha) = 6 Step 4: To find R, square both equations and add them: (R cos(alpha))^2 + (R sin(alpha))^2 = 8^2 + 6^2 R^2 cos^2(alpha) + R^2 sin^2(alpha) = 64 + 36 R^2 (cos^2(alpha) + sin^2(alpha)) = 100 Step 5: Use the Pythagorean identity cos^2(alpha) + sin^2(alpha) = 1: R^2 * 1 = 100 R^2 = 100 Step 6: Since R is an amplitude (positive value), we take the positive square root: R = sqrt(100) = 10 The amplitude R is 10.

6. tan(285°) = ? Answer: -(2+√3) Solution: Express 285° as a sum: 285° = 240° + 45°. Find exact values: tan(240°) = tan(180°+60°) = tan(60°) = √3 (since tangent has period 180°). tan(45°) = 1.
   Full step-by-step solution

   Step 1: Express 285° as a sum: 285° = 240° + 45°. Step 2: Apply the tangent sum formula: tan(A+B) = (tan A + tan B) / (1 - tan A tan B), with A = 240°, B = 45°. Step 3: Find exact values: tan(240°) = tan(180°+60°) = tan(60°) = √3 (since tangent has period 180°). tan(45°) = 1. Step 4: Substitute into the formula: tan(285°) = (√3 + 1) / (1 - √3 * 1) = (√3 + 1) / (1 - √3). Step 5: Rationalize the denominator: Multiply numerator and denominator by the conjugate (1 + √3): = (√3 + 1)(1 + √3) / ((1 - √3)(1 + √3)) = (√3 + 1)(√3 + 1) / (1 - 3) = ( (√3)^2 + 2√3 + 1 ) / (-2) = (3 + 2√3 + 1) / (-2) = (4 + 2√3) / (-2) = -2 - √3. The exact value of tan(285°) is -(2+√3).