Problem 18 - Precalculus: A Concise Course

Question

Evaluate (if possible) the sine, cosine, and tangent of the real number. $$ t=\frac{\pi}{3} $$

Step-by-Step Solution

Answer

The sine of \( t = \frac{\pi}{3} \) is \( \sqrt{3}/2 \), the cosine is \( 1/2 \), and the tangent is \( \sqrt{3} \).

Step 1: Calculate Sine

Sine of \( t = \frac{\pi}{3} \) can be directly calculated using the unit circle or the table of exact values: \( \sin(\frac{\pi}{3}) = \sqrt{3}/2 \)

Step 2: Calculate Cosine

Cosine of \( t = \frac{\pi}{3} \) can be directly calculated using the unit circle or the table of exact values: \( \cos(\frac{\pi}{3}) = 1/2 \)

Step 3: Calculate Tangent

The tangent of a number is given by the ratio of the sine to the cosine: \( \tan(t) = \sin(t)/\cos(t) \). So the tangent of \( t = \frac{\pi}{3} \) is \( \tan(\frac{\pi}{3}) = \sin(\frac{\pi}{3})/\cos(\frac{\pi}{3}) = (\sqrt{3}/2) / (1/2) = \sqrt{3} \)

Key Concepts

Unit Circle

The unit circle is a powerful tool that helps us understand trigonometric functions. Picture a circle centered at the origin of a coordinate plane - its radius is exactly 1 unit. This simple circle holds profound connections to sine, cosine, and tangent values, especially those termed 'exact values'.

On the unit circle, each angle corresponds to a unique point. For an angle \( t \), the x-coordinate of this point gives the cosine value, and the y-coordinate gives the sine value. Why is this helpful? Because these coordinates directly relate to a circle of radius 1, making the math simpler and often exact.

For \( t = \frac{\pi}{3} \), this angle corresponds to the exact coordinates \((1/2, \sqrt{3}/2)\). This gives us immediate sine and cosine values, showing just how handy the unit circle can be.

Exact Values

Exact values in trigonometry are specific values of trigonometric functions at particular angles that can be precisely expressed using square roots, fractions, or straightforward numbers. They are critical to solving many math problems exactly, rather than numerically or with approximations.

Angles like \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \) are commonly found on the unit circle, corresponding to exact trigonometric values. For example, \( \sin(\frac{\pi}{3}) = \sqrt{3}/2 \) and \( \cos(\frac{\pi}{3}) = 1/2 \). Recognizing these values quickly aids in solving equations and understanding relations between angles.

Memorizing these values can seem daunting, but with consistent practice, they'll become intuitive, like recognizing a close friend.

Tangent Ratio

The tangent function is an essential trigonometric function derived from sine and cosine. It is not directly depicted on the unit circle, unlike sine and cosine. Instead, it uses the tangent ratio, defined as \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). By relating sine and cosine in this way, tangent provides a different perspective on angles and their related functions.

For the angle \( t = \frac{\pi}{3} \), considering our knowledge of exact values, we have:

• \( \sin(\frac{\pi}{3}) = \sqrt{3}/2 \)
• \( \cos(\frac{\pi}{3}) = 1/2 \)

This calculation yields the tangent as \( \tan(\frac{\pi}{3}) = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \).

Understanding the tangent ratio is crucial for tackling many trigonometric problems, blending familiarity with the sine and cosine functions to offer a broad application of angles beyond just the circle.