How to find the exact value of cosine 5pi/12?

The exact value of cosine 5π/12 can be determined using trigonometric identities and properties. This article will guide you through the steps to find the precise value of cosine 5π/12 and provide answers to related frequently asked questions.

Solving for Cosine 5π/12

To find the exact value of cosine 5π/12, we need to apply trigonometric identities and formulas. Let’s break down the process step by step:

1. Start with the identity: cos(θ + φ) = cos(θ)cos(φ) – sin(θ)sin(φ).
2. Replace θ with 5π/12 and φ with 7π/12 to obtain: cos(5π/12 + 7π/12) = cos(5π/12)cos(7π/12) – sin(5π/12)sin(7π/12).
3. Simplify the equation using known values: cos(π) = -1 and sin(π) = 0.
4. The equation now becomes: cos(12π/12) = cos(5π/12) * (-sin(7π/12)).
5. Since cos(12π/12) = cos(π) = -1, we can rewrite the equation as: -1 = cos(5π/12) * (-sin(7π/12)).
6. Divide both sides of the equation by -sin(7π/12): cos(5π/12) = -1 / -sin(7π/12).
7. Simplify further: cos(5π/12) = 1 / sin(7π/12).
8. To solve for sin(7π/12), we use the identity: sin(θ) = sin(π – θ).
9. Applying this identity to sin(7π/12), we get sin(7π/12) = sin(π – 7π/12) = sin(π/12).
10. Our equation now becomes: cos(5π/12) = 1 / sin(π/12).
11. Finally, we can evaluate sin(π/12) using the exact value: sin(π/12) = √[(3 – √5) / 8], which can be derived through various trigonometric methods.
12. Substitute the value of sin(π/12) into our equation: cos(5π/12) = 1 / √[(3 – √5) / 8].

Frequently Asked Questions

1. How do I find the exact value of cosine?

To find the exact value of cosine, you need to use trigonometric identities and formulas specific to the angle given.

2. What is the cosine of 0 degrees?

The cosine of 0 degrees is 1.

3. How can I calculate cosine using a calculator?

Most scientific calculators have a cosine function represented as “cos.” Simply input the angle in radians or degrees, and the calculator will provide the cosine value.

4. What are the values of cosine for angles in the unit circle?

In the unit circle, cosine values vary from 1 (at 0 degrees) to -1 (at 180 degrees or π radians).

5. How do I convert degrees to radians?

To convert degrees to radians, multiply the degree measure by π/180.

6. Can the exact value of cosine be a fraction?

Yes, the exact value of cosine can sometimes be expressed as a fraction or radical.

7. What is the cosine of a negative angle?

The cosine of a negative angle is the same as the cosine of its positive counterpart.

8. How do I find the cosine of a sum or difference angle?

Use the cosine sum or difference formulas: cos(θ ± φ) = cos(θ)cos(φ) – sin(θ)sin(φ).

9. Can the exact value of cosine be irrational?

Yes, the exact value of cosine can be irrational, such as in the case of π/6.

10. Is it possible to determine cosine without using trigonometric identities?

Trigonometric identities and formulas are essential to finding exact values of cosine in most cases.

11. What is the cosine of 90 degrees?

The cosine of 90 degrees is 0.

12. How do I find the cosine of an obtuse angle?

To find the cosine of an obtuse angle, consider its reference angle (acute angle formed by the terminal side and the x-axis) and apply cos(θ) = ± cos(reference angle), depending on the quadrant.

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