Problem 63 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ \frac{\tan \frac{\pi}{3}}{2}-\frac{1}{\sec \frac{\pi}{6}} $$

Step-by-Step Solution

Answer

The exact value of the expression is \( \frac{\sqrt{3}-1}{2} \).

Step 1: Break down the given expression

The given expression is \(\frac{\tan \frac{\pi}{3}}{2}-\frac{1}{\sec \frac{\pi}{6}}\). It's necessary to break this into smaller units for easier computation. It becomes \(\frac{\tan \frac{\pi}{3}}{2}\) minus \(\frac{1}{\sec \frac{\pi}{6}}\)

Step 2: Compute for tan(pi/3)

From the unit circle, we can recall that the value of \( \tan \frac{\pi}{3} \) is \( \sqrt{3} \)

Step 3: Compute for sec(pi/6)

From the unit circle, we can recall that the value of \( \sec \frac{\pi}{6} \) is 2

Step 4: Compute for the expression

Substitute the values we know into the original expression. You get \(\frac{\sqrt{3}}{2}-\frac{1}{2}\). The two terms both have 2 as a denominator. Compute \( \frac{\sqrt{3}-1}{2} \).

Key Concepts

Unit circle

The unit circle is a vital tool in trigonometry and it helps us understand trigonometric functions, such as sine, cosine, and tangent, using geometric principles. Imagine a circle with a radius of 1 centered at the origin of a coordinate plane. This circle is known as the unit circle.
The angle in question is usually measured from the positive x-axis in a counterclockwise direction. The different points around the circumference of the circle represent angles, and each point provides the cosine and sine of those angles as the x and y coordinates, respectively. This becomes the foundation for identifying trigonometric values for these angles.

• On the unit circle, cosine is the x-coordinate.
• Sine is the y-coordinate.
• Tangent is the ratio of sine over cosine.

Recognizing these properties can simplify trigonometric calculations without needing a calculator, doing so by referencing basic angle values that are frequently used. The benefits are substantial because it avoids complex numerical calculations, making the unit circle an intuitive and direct way to find trigonometric values.

Exact values

Trigonometric functions have specific values at certain angles that are often referred to as 'exact values.' These include key angles such as \( \frac{\pi}{6} \, (30^\circ), \frac{\pi}{4} \, (45^\circ), \frac{\pi}{3} \, (60^\circ) \), and \( \frac{\pi}{2} \, (90^\circ) \).These angles and their corresponding trigonometric values are straightforward to memorize and use, given their common appearance in many mathematical problems.
For example, for \( \tan \frac{\pi}{3} \), the value is known to be \( \sqrt{3} \). Similarly, the secant function, which is the reciprocal of cosine, at angle \( \frac{\pi}{6} \) gives an exact value of 2. Using these precise values, we can quickly solve trigonometric equations such as the one in the exercise by substituting these known values into the expression.

Tangent and Secant

In trigonometry, the tangent and secant functions are among the six fundamental trigonometric functions and have unique roles. Let's dive into what they mean and how they're used:

• Tangent (tan): This function expresses the ratio of the sine to the cosine of an angle. It can be used to measure slope and inclination in a geometric context. For example, in our exercise, we used \( \tan \frac{\pi}{3} \) which gives \( \sqrt{3} \), indicating a particular steep slope.
• Secant (sec): The secant function is the reciprocal of the cosine function. It's particularly useful for deriving lengths of line segments in trigonometry. For instance, secant \( \sec \frac{\pi}{6} \) is 2 because it is the reciprocal of \( \cos \frac{\pi}{6} \), which is \( \frac{\sqrt{3}}{2} \).

Both of these functions can be used to simplify and solve trigonometric equations by employing their reciprocal and ratio properties. By practicing with these values and properties, it becomes simpler to comprehend and employ these functions effectively in various trigonometric problems.