**What is the exact value of cos 105?**
When it comes to evaluating trigonometric functions, especially those of angles that are not commonly found on the unit circle, it can be a bit tricky. In this case, we are interested in finding the exact value of cos 105.
To determine the value of cos 105, we need to understand a few key concepts and utilize certain trigonometric identities.
FAQs:
1. What is cosine?
Cosine (cos) is one of the trigonometric functions that relates the angles of a right triangle to the ratios of its sides.
2. What is the unit circle?
The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is commonly used in trigonometry to determine the values of trigonometric functions at various angles.
3. How can we find the exact value of cos 105?
To find the exact value of cos 105, we can use the angle sum formula for cosine. This formula states that cos(A + B) = cos A * cos B – sin A * sin B. In this case, we’ll use the angle A as 60 degrees and angle B as 45 degrees.
4. Why do we split cos 105 into cos 60 and cos 45?
By splitting cos 105 into cos 60 and cos 45, we can simplify the calculation since the trigonometric values for these angles are well-known.
5. What is the value of cos 60?
The value of cos 60 (cosine of 60 degrees) is 0.5. This is a readily available value on the unit circle.
6. What is the value of cos 45?
The value of cos 45 (cosine of 45 degrees) is (√2)/2, which is also a common trigonometric value.
7. How do we use the angle sum formula to find cos 105?
We can rewrite cos 105 as cos (60 + 45) and then apply the angle sum formula to get the exact value.
8. What are the trigonometric values for sin 60 and sin 45?
The value of sin 60 (sine of 60 degrees) is (√3)/2, and the value of sin 45 (sine of 45 degrees) is (√2)/2.
9. How do we evaluate the product cos 60 * cos 45?
To evaluate the product cos 60 * cos 45, we simply multiply the values obtained for each angle. Thus, (0.5) * (√2)/2 = (√2)/4.
10. How do we evaluate the product sin 60 * sin 45?
To evaluate the product sin 60 * sin 45, we multiply the sine values obtained for each angle. This results in (√3)/2 * (√2)/2 = (√6)/4.
11. What is the value of cos 105?
The exact value of cos 105 is obtained by plugging the values of cos 60, cos 45, sin 60, and sin 45 into the angle sum formula. Substituting the known values, we have cos 105 = cos 60 * cos 45 – sin 60 * sin 45 = (0.5)(√2)/2 – (√3)/2 * (√2)/2 = (√2)/4 – (√6)/4.
12. Can we simplify (√2)/4 – (√6)/4 further?
Yes, we can simplify (√2)/4 – (√6)/4 further by combining the terms with the same denominator. This simplifies to (√2 – √6)/4.
In conclusion, the exact value of cos 105 is (√2 – √6)/4. By applying the angle sum formula and utilizing the trigonometric values of cos 60 and cos 45, we can obtain the precise value of this trigonometric function.