A triangle is inscribed in a circle with radius 8 cm. Two sides of the triangle measure 12 cm and 10 cm, and the angle between them is 60°. Using the Law of Cosines, find the length of the third side of the triangle.Answer: ______________
Emma is analyzing sound waves in her physics class and needs to find the exact value of sin(75°) for a wave interference calculation. Using sum and difference formulas, determine the exact value of sin(75°).Answer: ______________
Liam is designing a suspension bridge and needs to calculate the exact height of the main cable at a specific point. The cable follows the path y = 50sin(πx/200) where x is the horizontal distance from the left tower in meters. Using the sum-to-product identities, determine the exact height of the cable when x = 50 meters.Answer: ______________
Emma is analyzing sound waves in her physics class and needs to find the exact value of sin(75°)cos(15°) - cos(75°)sin(15°). Using trigonometric identities, what is the exact value of this expression?Answer: ______________
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: ______________
A triangle is inscribed in a circle with radius 8 cm. Two sides of the triangle measure 12 cm and 10 cm, and the angle between them is 60°. Using the Law of Cosines, find the length of the third side of the triangle.Answer: 2√37 cm Solution: The Law of Cosines is a generalization of the Pythagorean theorem that applies to any triangle, not just right triangles.Full step-by-step solution
The Law of Cosines is a generalization of the Pythagorean theorem that applies to any triangle, not just right triangles. It states that the square of one side equals the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. This allows you to find unknown sides when you know two sides and the angle between them. The circle's radius in this problem provides context but isn't needed for the calculation using the Law of Cosines.
Emma is analyzing sound waves in her physics class and needs to find the exact value of sin(75°) for a wave interference calculation. Using sum and difference formulas, determine the exact value of sin(75°).Answer: (√6 + √2)/4 Solution: Express 75° as a sum of two angles with known trigonometric values: 75° = 45° + 30° Apply the sine sum formula: sin(A + B) = sinA cosB + cosA sinB Substitute A = 45° and B = 30°: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°) Use exact values: sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2,…Full step-by-step solution
Step 1: Express 75° as a sum of two angles with known trigonometric values: 75° = 45° + 30°
Step 2: Apply the sine sum formula: sin(A + B) = sinA cosB + cosA sinB
Step 3: Substitute A = 45° and B = 30°: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)
Step 4: Use exact values: sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2, sin(30°) = 1/2
Step 5: Substitute the values: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)
Step 6: Simplify: sin(75°) = (√6/4) + (√2/4)
Step 7: Combine terms: sin(75°) = (√6 + √2)/4
The answer is (√6 + √2)/4.
Liam is designing a suspension bridge and needs to calculate the exact height of the main cable at a specific point. The cable follows the path y = 50sin(πx/200) where x is the horizontal distance from the left tower in meters. Using the sum-to-product identities, determine the exact height of the cable when x = 50 meters.Answer: 25√2 Solution: We are given the cable path: y = 50 sin(πx / 200). We need the height at x = 50 meters. Substitute x = 50 into the equation.Full step-by-step solution
We are given the cable path: y = 50 sin(πx / 200).
We need the height at x = 50 meters.
Step 1: Substitute x = 50 into the equation.
y = 50 sin(π * 50 / 200)
Step 2: Simplify the argument of the sine.
π * 50 / 200 = π * (50 / 200) = π * (1 / 4) = π/4.
So y = 50 sin(π/4).
Step 3: Recall the exact value of sin(π/4).
sin(π/4) = √2 / 2.
Step 4: Substitute that value.
y = 50 * (√2 / 2)
Step 5: Simplify.
50 * √2 / 2 = (50 / 2) * √2 = 25 * √2.
So the exact height is 25√2 meters.
Final answer: 25√2
Emma is analyzing sound waves in her physics class and needs to find the exact value of sin(75°)cos(15°) - cos(75°)sin(15°). Using trigonometric identities, what is the exact value of this expression?Answer: 1/2 Solution: Recognize that the expression sin(75°)cos(15°) - cos(75°)sin(15°) matches the sine difference formula: sin(A - B) = sinAcosB - cosAsinB Identify A = 75° and B = 15° Apply the formula: sin(75° - 15°) = sin(60°) Calculate sin(60°) = √3/2 The exact value is √3/2 The answer is √3/2.Full step-by-step solution
Step 1: Recognize that the expression sin(75°)cos(15°) - cos(75°)sin(15°) matches the sine difference formula: sin(A - B) = sinAcosB - cosAsinB
Step 2: Identify A = 75° and B = 15°
Step 3: Apply the formula: sin(75° - 15°) = sin(60°)
Step 4: Calculate sin(60°) = √3/2
Step 5: The exact value is √3/2
The answer is √3/2.
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.
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 √3/2Full 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
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