Trigonometric Identities and Equations

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Q1. Which equation represents the double angle identity for sine?

Solution:

The double angle identity for sine is sin(2x) = 2sin(x)cos(x).

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Q2. Which of the following is a Pythagorean identity?

Solution:

The Pythagorean identity is sin²x + cos²x = 1.

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Q3. Which of the following is the correct identity for cotangent?

Solution:

cot(x) = cos(x)/sin(x) is the correct identity.

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Q4. Solve for x: cos(x) = 0.5, where 0 ≤ x < 360°.

Solution:

cos(x) = 0.5 gives x = 60° and 300°.

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Q5. What is the value of cos(45°)?

Solution:

cos(45°) = √2/2.

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Q6. Which of the following is the correct expression for cot(x)?

Solution:

cot(x) = 1/tan(x) = cos(x)/sin(x).

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Q7. Solve for x: 2sin(x) = √3, where 0 ≤ x < 360°.

Solution:

sin(x) = √3/2 gives x = 60° and 120°, so 2sin(x) = √3 gives x = 150°.

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Q8. If tan(x) = 1, what is the value of x in the interval [0°, 360°)?

Solution:

tan(x) = 1 at x = 45° and 225°; in the interval [0°, 360°), the first solution is 45°.

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Q9. If sin(x) = 0.5, what are the possible values of x in degrees?

Solution:

sin(x) = 0.5 gives x = 30° and 150°.

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Q10. If sin(x) = 0.8, what is the value of csc(x)?

Solution:

csc(x) = 1/sin(x) = 1/0.8 = 1.25.