Problem 66 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ \frac{10 \pi}{3} $$

Step-by-Step Solution

Answer

Therefore, for the angle \(\frac{10 \pi}{3}\), the sine is \(\frac{\sqrt{3}}{2}\), the cosine is \(\frac{1}{2}\), and the tangent is \(\sqrt{3}\)

Step 1: Simplify The Input Angle

Simplify the angle \(\frac{10 \pi}{3}\). It can be thought of as \(3\pi + \frac{\pi}{3}\). The value \(3\pi\) falls outside the range of a single period, \(2\pi\). Hence we can omit this and just take into calculation the fractional part \(\frac{\pi}{3}\). Therefore, trigonometric values for \(\frac{10 \pi}{3}\) would be the same as for \(\frac{\pi}{3}\)

Step 2: Find Sine, Cosine, and Tangent Values

Find the sine, cosine, and tangent for \(\frac{\pi}{3}\). Recall that \(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\), \(\cos(\frac{\pi}{3})=\frac{1}{2}\), and \(\tan(\frac{\pi}{3})=\sqrt{3}\)

Key Concepts

Sine

The sine function is a fundamental concept in trigonometry. It helps us understand the relationship between angles and side lengths in a right triangle. Specifically, for an angle \(\theta\), the sine function relates the length of the side opposite \(\theta\) to the hypotenuse. This is expressed as \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
In the context of a circle, sine represents a vertical distance from the origin.

• For example, for the angle \(\frac{\pi}{3}\), you can imagine a right triangle inside the unit circle.
• The opposite side there would then have a length of \(\sqrt{3}/2\).

Thus, \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\).
Remember that the sine function, like other trigonometric functions, is periodic with a period of \(2\pi\). This means the values repeat every \(2\pi\) units.

Cosine

Cosine is another important trigonometric function that helps in relating angles and side lengths in right triangles. For an angle \(\theta\), cosine defines the ratio of the length of the adjacent side to the hypotenuse. This is expressed as \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
In the unit circle, cosine represents a horizontal distance from the origin.

• When dealing with an angle \(\frac{\pi}{3}\), consider the adjacent side inside the unit circle which measures \(1/2\).

Hence, \(\cos(\frac{\pi}{3}) = \frac{1}{2}\).
Like the sine function, the cosine function is also periodic with a period of \(2\pi\). This repeating behavior can help in simplifying calculations for angles outside one full rotation.

Tangent

The tangent function gives insight into the relationship between trigonometric functions. It is obtained by taking the ratio of the sine function to the cosine function for a given angle \(\theta\). This is expressed as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
The tangent represents the slope of the line created with the angle in the coordinate plane.

• For \(\frac{\pi}{3}\), with \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\) and \(\cos(\frac{\pi}{3}) = \frac{1}{2}\), tangent is calculated by dividing the sine value by the cosine value.
• This calculation leads to \(\tan(\frac{\pi}{3}) = \sqrt{3}\).

The tangent function, like sine and cosine, is periodic, but with a different period of \(\pi\). Understanding these relationships helps in evaluating trigonometric expressions without a calculator.