Question

Find the exact value of each expression. Do not use a calculator. $$ \tan (-6 \pi)+\cos \frac{9 \pi}{4} $$

Short answer

\[ \frac{\frac{\frac{\sqrt{2}}{2} }{\sqrt{2}}{2} }{\sqrt{2}}{2} \]

Step-by-step solution

Simplify the tangent expression

Use the periodic property of the tangent function. Recall that the period of \( \tan(x) \) is \( \pi \), meaning \( \tan(x + k \pi) = \tan(x) \) for any integer \( k \). Thus, \( \tan(-6 \pi) \) simplifies to \( \tan(0) \).

Evaluate the simplified tangent expression

Since \( \tan(0) = 0 \), we find that \( \tan(-6 \pi) = 0 \).

Simplify the cosine expression

Use the periodic property of the cosine function. Recall that the period of \( \cos(x) \) is \( 2 \pi \), meaning \( \cos(x + 2 k \pi) = \cos(x) \) for any integer \( k \). Thus, \( \cos \left( \frac{9 \pi}{4} \right) = \cos \left( \frac{9 \pi}{4} - 2 \pi \right) = \cos \left( \frac{9 \pi}{4} - \frac{8 \pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) \).

Evaluate the simplified cosine expression

Now, since \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \), we find that \( \cos \left( \frac{9 \pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Combine results

Finally, combine the results of the simplified expressions. Thus, \( \tan(-6 \pi) + \cos \left( \frac{9 \pi}{4} \right) = 0 + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \).

Key concepts

periodic properties

Trigonometric functions like sine, cosine, and tangent are periodic. This means they repeat their values in regular intervals. For the cosine function, this period is \(2\pi\), and for the tangent function, it is \(\pi\). This property allows us to simplify complex expressions.
For example, in our exercise, the expression \(\tan(-6 \pi)\) can be simplified because tangent repeats every \(\pi\). Therefore, \(\tan(x) = \tan(x + k\pi)\) for any integer \(k\). Similarly, the expression \(\cos\left( \frac{9\pi}{4} \right)\) can be simplified using the cosine's periodic property. Since cosine repeats every \(2\pi\), \(\cos(x) = \cos(x + 2k\pi)\) for any integer \(k\).
Understanding periodic properties is crucial in solving trigonometric problems efficiently.

tangent function

The tangent function, tan(x), is one of the primary trigonometric functions. It can be defined as the ratio of the sine and cosine functions: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).
Tangent is periodic, with a period of \(\pi\). This feature means the function repeats its values every \(\pi\) units along the x-axis. For example, \(\tan(0) = 0\) and \(\tan(\pi) = 0\), illustrating the repeating nature.
In our exercise, \(\tan(-6 \pi)\) simplifies to \(\tan(0)\) because of the periodic property of the tangent function. Since \(\tan(0) = 0\), we find \(\tan(-6 \pi) = 0\).

cosine function

The cosine function, cos(x), is another fundamental trigonometric function. It represents the x-coordinate of a point on the unit circle as one moves counterclockwise from the positive x-axis.
Cosine is periodic with a period of \(2\pi\), meaning \(\cos(x) = \cos(x + 2k\pi)\) for any integer \(k\). This property helps simplify expressions like \(\cos\left( \frac{9\pi}{4} \right)\).
In our exercise, we reduced \(\cos\left( \frac{9\pi}{4} \right)\) to \(\cos\left( \frac{\pi}{4} \right)\) using the periodic property. We know that \(\cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\). Hence, \(\cos\left( \frac{9\pi}{4} \right) = \frac{\sqrt{2}}{2}\).

exact values

Exact values are critical in trigonometry because they allow us to solve problems without relying on calculators. Several trigonometric values are well-known and commonly used.
For instance, we know the exact value of \(\cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\) and \(\tan(0) = 0\). These values are derived from the unit circle and the definition of trigonometric functions.
In our exercise, combining these exact values gave us the final result. We found \(\tan(-6 \pi) + \cos\left( \frac{9\pi}{4} \right) = 0 + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}\). Mastering these exact values is essential for tackling various trigonometric problems efficiently.