Rewriting Trigonometric Expressions Exact Values Without Calculator

In this article, we will delve into the process of rewriting trigonometric expressions with angles within the range of $0 ext{ to } 2 ext{\pi}$ and then proceed to determine the exact values of these expressions without resorting to a calculator. This involves understanding the periodicity of trigonometric functions, reference angles, and the unit circle. We will tackle three specific examples: $\cos(\frac{42\pi}{8})$, $\csc(\frac{13\pi}{3})$, and $\tan(-\frac{17\pi}{6})$. By working through these examples, we aim to solidify the methodology and enhance comprehension of trigonometric manipulations.

a.) Rewriting and Evaluating $\cos(\frac{42\pi}{8})$

Let's begin by focusing on rewriting cosine functions. Our first task is to rewrite the expression $\cos(\frac{42\pi}{8})$ with an angle $\theta$ such that $0 \leq \theta < 2\pi$. To achieve this, we first simplify the fraction within the cosine function. $\frac{42\pi}{8}$ simplifies to $\frac{21\pi}{4}$. Now, we need to find an angle coterminal with $\frac{21\pi}{4}$ that lies within the desired interval. We can do this by subtracting multiples of $2\pi$ from $\frac{21\pi}{4}$ until we obtain an angle within the range of $0 \leq \theta < 2\pi$. Recall that $2\pi$ can be rewritten as $\frac{8\pi}{4}$.

Subtracting multiples of $\frac{8\pi}{4}$:

• $\frac{21\pi}{4} - \frac{8\pi}{4} = \frac{13\pi}{4}$
• $\frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}$

Since $\frac{5\pi}{4}$ falls within the range $0 \leq \theta < 2\pi$, we can rewrite the original expression as:

$\cos(\frac{42\pi}{8}) = \cos(\frac{5\pi}{4})$

Now, to find the exact value, we need to determine the reference angle for $\frac{5\pi}{4}$. The angle $\frac{5\pi}{4}$ lies in the third quadrant, where both the x and y coordinates are negative. To find the reference angle, we subtract $\pi$ from $\frac{5\pi}{4}$:

Reference angle = $\frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4}$

The reference angle is $\frac{\pi}{4}$, which corresponds to 45 degrees. We know that $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$. However, since $\frac{5\pi}{4}$ is in the third quadrant where cosine is negative, we have:

$\cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$

Therefore, the exact value of $\cos(\frac{42\pi}{8})$ is $-{\frac{\sqrt{2}}{2}}$. This process highlights the importance of understanding coterminal angles and reference angles in evaluating trigonometric functions.

b.) Rewriting and Evaluating $\csc(\frac{13\pi}{3})$

Next, let's turn our attention to the cosecant function. Our goal here is to rewrite $\csc(\frac{13\pi}{3})$ with an angle within the standard range and then find its exact value. Similar to the cosine example, we begin by finding a coterminal angle within the range of $0 \leq \theta < 2\pi$. We subtract multiples of $2\pi$ (or $\frac{6\pi}{3}$) from $\frac{13\pi}{3}$:

• $\frac{13\pi}{3} - \frac{6\pi}{3} = \frac{7\pi}{3}$
• $\frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}$

Thus, $\frac{\pi}{3}$ is coterminal with $\frac{13\pi}{3}$ and falls within the desired range. We can rewrite the expression as:

$\csc(\frac{13\pi}{3}) = \csc(\frac{\pi}{3})$

To evaluate the cosecant function, we need to recall that cosecant is the reciprocal of the sine function: $\csc(\theta) = \frac{1}{\sin(\theta)}$. Therefore, we need to find the value of $\sin(\frac{\pi}{3})$.

The angle $\frac{\pi}{3}$ corresponds to 60 degrees, which is a special angle. We know that $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$. Consequently:

$\csc(\frac{\pi}{3}) = \frac{1}{\sin(\frac{\pi}{3})} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{3}$:

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Therefore, the exact value of $\csc(\frac{13\pi}{3})$ is $\frac{2\sqrt{3}}{3}$. This example reinforces the importance of recognizing reciprocal trigonometric functions and using special angle values.

c.) Rewriting and Evaluating $\tan(-\frac{17\pi}{6})$

Finally, let's consider the tangent function. We are tasked with rewriting $\tan(-\frac{17\pi}{6})$ and finding its exact value. Since the angle is negative, we need to add multiples of $2\pi$ (or $\frac{12\pi}{6}$) until we obtain an angle within the range of $0 \leq \theta < 2\pi$:

• $-\frac{17\pi}{6} + \frac{12\pi}{6} = -\frac{5\pi}{6}$
• $-\frac{5\pi}{6} + \frac{12\pi}{6} = \frac{7\pi}{6}$

Thus, $\frac{7\pi}{6}$ is coterminal with $-\frac{17\pi}{6}$ and falls within the desired range. We can rewrite the expression as:

$\tan(-\frac{17\pi}{6}) = \tan(\frac{7\pi}{6})$

To find the exact value of $\tan(\frac{7\pi}{6})$, we need to determine the reference angle. The angle $\frac{7\pi}{6}$ lies in the third quadrant, where tangent is positive. The reference angle is found by subtracting $\pi$ from $\frac{7\pi}{6}$:

Reference angle = $\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}$

The reference angle is $\frac{\pi}{6}$, which corresponds to 30 degrees. We know that $\tan(\frac{\pi}{6}) = \frac{\sqrt{3}}{3}$. Since $\frac{7\pi}{6}$ is in the third quadrant, where tangent is positive, we have:

$\tan(\frac{7\pi}{6}) = \frac{\sqrt{3}}{3}$

Therefore, the exact value of $\tan(-\frac{17\pi}{6})$ is $\frac{\sqrt{3}}{3}$. This example illustrates how to handle negative angles and reinforces the importance of quadrant awareness when determining the sign of trigonometric functions.

Conclusion

In summary, we have successfully rewritten the trigonometric expressions $\cos(\frac{42\pi}{8})$, $\csc(\frac{13\pi}{3})$, and $\tan(-\frac{17\pi}{6})$ with angles between $0$ and $2\pi$ and then found their exact values without the aid of a calculator. This process involved simplifying fractions, finding coterminal angles, determining reference angles, and applying knowledge of special angle values and quadrant signs. Mastering these techniques is crucial for a strong understanding of trigonometry and its applications in various fields.

Key takeaways from this article:

• Coterminal angles: Finding coterminal angles within the range of $0 \leq \theta < 2\pi$ is crucial for simplifying trigonometric expressions. This involves adding or subtracting multiples of $2\pi$ until the angle falls within the desired range.
• Reference angles: Reference angles are the acute angles formed between the terminal side of an angle and the x-axis. They help determine the trigonometric values in different quadrants. Understanding reference angles simplifies the process of finding exact values for angles outside the first quadrant.
• Special angles: Knowing the trigonometric values for special angles such as $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$ is essential for evaluating trigonometric expressions without a calculator.
• Quadrant signs: The sign of a trigonometric function depends on the quadrant in which the angle lies. Remembering the mnemonic