Question

Find the exact value of each expression. Do not use a calculator. $$ \sec \frac{\pi}{6}+2 \csc \frac{\pi}{4} $$

Short answer

\[ \frac{2\text{√}3}{3} + 2\text{√}2 \]

Step-by-step solution

Identify trigonometric functions involved

The expression includes \(\sec \frac{\pi}{6}\) and \(\csc \frac{\pi}{4}\). We need to determine the exact values of these trigonometric functions.

Find \(\sec \frac{\pi}{6}\) value

The secant function is the reciprocal of the cosine function. Therefore, \(\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}}\). Knowing that \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), we get: \[\sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\].

Find \(\csc \frac{\pi}{4}\) value

The cosecant function is the reciprocal of the sine function. Therefore, \(\csc \frac{\pi}{4} = \frac{1}{\sin \frac{\pi}{4}}\). Knowing that \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\), we get: \[\csc \frac{\pi}{4} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}\].

Substitute the values into the expression

The expression is \(\sec \frac{\pi}{6} + 2 \csc \frac{\pi}{4}\). Substitute the values found: \(\frac{2\sqrt{3}}{3} + 2 \cdot \sqrt{2}\).

Simplify the expression

Combine the terms: \(\frac{2\sqrt{3}}{3} + 2\sqrt{2}\). This is the final exact value of the expression.

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where the functions are defined. They provide a way to simplify and solve trigonometric expressions.
There are several key identities in trigonometry:

• Pythagorean Identities: These relate the squares of the sine and cosine functions. For example, the identity \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\) is widely used.
• Reciprocal Identities: These show the relationships between one trigonometric function and the reciprocal of another. For instance, \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\) and \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\).
• Angle Sum and Difference Identities: These are used to find the trigonometric functions of angle sums and differences. For example, \(\text{sin}(a \pm b) = \text{sin} a \text{cos} b \pm \text{cos} a \text{sin} b\).

Understanding these identities is fundamental in solving trigonometric problems, including finding the exact values of expressions as seen in the given exercise.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions at specific angles involves knowing the values of these functions at key angles like \( \frac{\text{π}}{6}\), \( \frac{\text{π}}{4}\), and \( \frac{\text{π}}{3}\). You don't need a calculator if you remember these values.

• For example, the cosine and secant of \( \frac{\text{π}}{6}\): \(\text{cos} \frac{\text{π}}{6} = \frac{\text{√3}}{2}\) and \(\text{sec} \frac{\text{π}}{6} = \frac{1}{\text{cos} \frac{\text{π}}{6}} = \frac{2}{\text{√3}}\). By rationalizing, we get \(\text{sec} \frac{\text{π}}{6} = \frac{2\text{√3}}{3}\).
• For sine and cosecant of \(\frac{\text{π}}{4}\): \( \text{sin} \text{\frac{π}}{4} = \text{\frac{√2}}{2}\) and \( \text{csc} \frac{\text{π}}{4} = \text{\frac{1}}{\text{sin} \text{\frac{π}}{4}} = \text{\frac{2}}{\text{√2}} = \text{√2}\).

These exact values are handy when solving trigonometric expressions like \( \frac{2\text{√3}}{3} + 2\text{√2}\). By substituting the exact values, we simplify our calculations.

Secant and Cosecant Functions

Secant (\text{sec}) and cosecant (\text{csc}) are the reciprocal trigonometric functions of cosine (\text{cos}) and sine (\text{sin}) respectively.
Understanding their relationships:

• Secant is defined as \( \text{sec} θ = \frac{1}{\text{cos} θ} \): It's the reciprocal of cosine. For example, if \( \text{cos} \text{\frac{π}}{6} = \frac{\text{√3}}{2}\), then \( \text{sec} \text{\frac{π}}{6} = \frac{2}{\text{√3}} = \frac{2\text{√3}}{3}\).
• Cosecant is defined as \( \text{csc} θ = \frac{1}{\text{sin} θ} \): It's the reciprocal of sine. For example, if \( \text{sin} \text{\frac{π}}{4} = \frac{\text{√2}}{2}\), then \( \text{csc} \text{\frac{π}}{4} = \frac{2}{\text{√2}} = \text{√2}\).

It is essential to learn these reciprocal relationships because they can quite often simplify the process of solving trigonometric expressions. For example, in our given problem, converting \( \text{sec} \frac{\text{π}}{6} \) and \( \text{csc} \frac{\text{π}}{4}\) into their exact values lets us solve it without reaching for a calculator.