Problem 20 - Algebra 2

Question

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(\sin \theta=\cos \theta\)

Step-by-Step Solution

Answer

\( \theta = 45^\circ + 180^\circ k \), where \( k \) is an integer.

Step 1: Understanding the given equation

We are given the equation \( \sin \theta = \cos \theta \). Our goal is to find all values of \( \theta \) in degrees that satisfy this equation.

Step 2: Expressing using tangent

Recall that \( \sin \theta = \cos \theta \) is equivalent to \( \frac{\sin \theta}{\cos \theta} = 1 \). This fraction is the definition of tangent, so we have \( \tan \theta = 1 \).

Step 3: Solving for theta

We solve \( \tan \theta = 1 \). \( \tan \theta = 1 \) occurs at specific angles, known as the solutions of \( \theta = 45^\circ + 180^\circ k \). Here, \( k \) is any integer, representing the periodic nature of the tangent function.

Step 4: General Solution

Therefore, the general solution for \( \theta \) is \( \theta = 45^\circ + 180^\circ k \), where \( k \) is an integer, capturing all solutions in degrees where the sine and cosine functions are equal.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions. It plays a crucial role in relating the sine and cosine of an angle. The tangent of an angle \( \theta \) is defined as the ratio of the sine of \( \theta \) to the cosine of \( \theta \):

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

When \( \sin \theta \) equals \( \cos \theta \), dividing both sides by \( \cos \theta \) gives us \( \tan \theta = 1 \). Therefore, finding when \( \sin \theta = \cos \theta \) is the same as finding when \( \tan \theta = 1 \).
This equation simplifies the problem significantly and sets the stage for further solving. Tangent is unique from sine and cosine because it can take any real number value.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions over certain intervals. The tangent function is periodic with a period of \( 180^\circ \) (or \( \pi \) radians). This means every \( 180^\circ \), the tangent function produces the same value again.

• \( \tan(\theta + 180^\circ) = \tan \theta \)

Understanding periodicity is vital, as it helps us find multiple solutions to trigonometric equations, such as \( \tan \theta = 1 \), across the angles.
The periodic nature of tangent means that once you find one solution, you can find all others by adding or subtracting multiples of \( 180^\circ \). This property allows us to express solutions in a general form, accounting for all possible angle measures.

General Solution

A general solution provides a way to express all possible solutions of a trigonometric equation in one expression. For the equation \( \tan \theta = 1 \), we first find the principal solution, which is the simplest angle that satisfies the equation: \( \theta = 45^\circ \).
By acknowledging the periodicity of the tangent function, we can extend this to a general solution:

• \( \theta = 45^\circ + 180^\circ k \)

Here, \( k \) can be any integer, representing all angles where the original equation holds true. The term \( 180^\circ k \) accounts for the periodicity, ensuring that every angle where \( \tan \theta = 1 \) is covered. Thus, the general solution represents a comprehensive method to determine all solutions.

Degrees

Degrees are a unit of measure for angles in trigonometry, alongside radians. A full circle is \( 360^\circ \), and angles can be expressed in readings such as \( 45^\circ, 90^\circ, 180^\circ \), etc. In this exercise, we solve the equation \( \sin \theta = \cos \theta \) with angles measured in degrees, making it more accessible for applications such as navigation and geometry.

• A right angle is \( 90^\circ \).
• Straight angles measure \( 180^\circ \).
• Circular angles are \( 360^\circ \).

Measuring in degrees allows us to easily relate these solutions to everyday contexts where angles are expressed in this familiar unit. By using degrees, the problem becomes intuitive to solve, especially when setting the general solution \( \theta = 45^\circ + 180^\circ k \), which is naturally laid out in degrees.