What is the exact value of sin 75 degrees?
Understand the Problem
The question is asking for the exact value of the sine of 75 degrees, which can be found using trigonometric identities.
Answer
The exact value of $\sin(75^\circ)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$.
Answer for screen readers
The exact value of $\sin(75^\circ)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$.
Steps to Solve
- Use the sine angle addition formula
The sine of 75 degrees can be expressed as the sum of two angles:
$$ 75^\circ = 45^\circ + 30^\circ $$
Using the sine angle addition formula, we have:
$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) $$
The formula is:
$$ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) $$
- Substituting the values
Now substitute $a = 45^\circ$ and $b = 30^\circ$ into the formula:
$$ \sin(75^\circ) = \sin(45^\circ) \cos(30^\circ) + \cos(45^\circ) \sin(30^\circ $$
We know the values of the trigonometric functions:
- $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
- $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
- $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
- $\sin(30^\circ) = \frac{1}{2}$
Substituting these values gives:
$$ \sin(75^\circ) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) $$
- Simplifying the expression
Now simplify the expression:
$$ \sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$
Combine the terms:
$$ \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$
The exact value of $\sin(75^\circ)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$.
More Information
The sine of 75 degrees can also be useful in various applications, particularly in physics and engineering, where understanding angles and their relationships is crucial. Knowing how to express angles in terms of familiar values (like 30 degrees and 45 degrees) simplifies calculations enormously.
Tips
- Not using the correct angles for the sine addition formula.
- Forgetting to include both sine and cosine terms in the formula.
- Failing to simplify the final expression correctly.