Problem 98 - Algebra and Trigonometry

Question

Use an identity and not a calculator to find the value of each expression. a. \(\cos 47^{\circ} \sec 47^{\circ}\) b. \(\sin ^{2} \frac{\pi}{5}+\cos ^{2} \frac{\pi}{5}\)

Step-by-Step Solution

Answer

The answers are: a. 1 b. 1

Step 1: Understand The Identities

Firstly, let's start with understanding the identities being used in the exercise. The \( \sec \) function is defined as the reciprocal of the \( \cos \) function, so \( \sec(x) = 1/ \cos(x) \). This means that \( \cos(x). \sec(x) = 1 \). The second identity is the Pythagorean trigonometric identity, which states that \( \sin^2(x) + \cos^2(x) = 1 \) for any value of x.

Step 2: Apply the Identities - Part A

Applying the identity \( \cos(x). \sec(x) = 1 \) to the expression \( \cos(47^{\circ}) \sec(47^{\circ}) \), we get \( 1 \times 1 = 1 \). This is because the \( \cos \) and \( \sec \) of the same angle cancel each other out to give 1.

Step 3: Apply the Identities - Part B

The second expression \( \sin ^{2} \frac{\pi}{5} + \cos ^{2} \frac{\pi}{5} \) is the same as \( \sin^2(x) + \cos^2(x) \) for \( x = \frac{\pi}{5} \), which according to the Pythagorean trigonometric identity, equals 1.

Key Concepts

Reciprocal Trigonometric Functions

In trigonometry, reciprocal trigonometric functions are quite important. They are "reciprocal" because they are defined as the inverse of the basic trigonometric functions: sine, cosine, and tangent. Here, we focus on cosine and its reciprocal function, secant.

To understand this, remember the definition: the secant of an angle \( x \), represented as \( \sec(x) \), is the reciprocal of the cosine of that angle. Mathematically, this is expressed as:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Applying this concept means that when you multiply the cosine of an angle by the secant of the same angle, you get:

• \( \cos(x) \cdot \sec(x) = 1 \)

This happens because the \( \cos(x) \) and \( \sec(x) \) effectively "cancel" each other out, resulting in the value of 1. This principle can be very handy when solving trigonometric expressions, as it simplifies calculations significantly.

Pythagorean Identity

Pythagorean identity is one of the foundational elements of trigonometry. It's essential to keep this identity in mind, as it often simplifies expressions or helps verify calculations.

The identity itself is simple:

• \( \sin^2(x) + \cos^2(x) = 1 \)

This formula states that for any angle \( x \), the square of the sine of the angle plus the square of the cosine of the angle ALWAYS equals 1.

• The identity is named after the Pythagorean theorem because it resembles the relationship with the sides of a right triangle, where the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In practical terms, if you're given an angle and you know either \( \sin(x) \) or \( \cos(x) \), you can easily find the other value using this identity. It's a powerful tool that reinforces the interdependence of the sine and cosine functions in trigonometry.

Angle Measurement

Understanding angle measurements is crucial when working with trigonometry. Angles can be measured in degrees or radians. Both systems are used in different fields but knowing how to navigate between and understand both is important.

**Degrees**

• A degree is a unit of measurement that divides one full rotation into 360 parts. So, one complete circle is 360 degrees. For example, a right angle is 90 degrees.

**Radians**

• A radian is another way to measure angles, used mainly in advanced mathematics. One complete circle measures \( 2\pi \) radians. Thus, \( \pi \) radians equals 180 degrees. This system relates directly to the properties of circles, making it particularly useful in calculus.

Conversion between degrees and radians is a vital skill. Use the relationships:

• 1 degree = \( \frac{\pi}{180} \) radians
• 1 radian = \( \frac{180}{\pi} \) degrees

Whether you're examining a geometrical shape or solving trigonometric identities, understanding these units helps ensure your calculations are correct and meaningful.