cos 15 exact value
Understand the Problem
The question is asking for the exact value of cos(15 degrees). This involves using trigonometric identities or formulas to find the value without using a calculator.
Answer
$$ \frac{\sqrt{6} + \sqrt{2}}{4} $$
Answer for screen readers
The exact value of $\cos(15^\circ)$ is given by $$ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$
Steps to Solve
- Use the Cosine Difference Formula
To find $\cos(15^\circ)$, we can express it using a known angle: $$ \cos(15^\circ) = \cos(45^\circ - 30^\circ) $$ We will apply the cosine difference identity: $$ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) $$
- Substitute Known Values
Next, we substitute $a = 45^\circ$ and $b = 30^\circ$. So, $$ \cos(15^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ $$
- Calculate Each Trigonometric Function
Now we need the values of the trigonometric functions at these angles:
- $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
- $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
- $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
- $\sin(30^\circ) = \frac{1}{2}$
Plugging these values into our equation gives: $$ \cos(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$
- Combine the Terms
Now we simplify the expression: $$ \cos(15^\circ) = \frac{\sqrt{2}\sqrt{3}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} $$
The exact value of $\cos(15^\circ)$ is given by $$ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$
More Information
The cosine values for angles like 15 degrees can be derived using trigonometric identities such as the cosine difference formula. This is a useful method when calculators aren't available. Knowing these angles and their cosine values can also aid in solving more complex problems.
Tips
- A common mistake is to confuse the angles when applying the cosine difference formula. It's important to correctly identify the angles being used.
- Forgetting to apply the signs correctly when combining terms can lead to incorrect results. Pay attention to addition and ensure all terms are correctly calculated.