Problem 23 - Algebra 2 and Trigonometry

Question

If \(\sin \theta=\cos (20+\theta),\) what is the value of \(\theta ?\)

Step-by-Step Solution

Answer

The value of \(\theta\) is \(35^\circ\).

Step 1: Set up the equation

Start by writing down the given equation: \(\sin \theta = \cos (20+\theta)\). We can use the co-function identity, \(\cos(90^\circ - x) = \sin x\), in our solution.

Step 2: Apply the co-function identity

According to the co-function identity, we have \(\cos (20 + \theta) = \cos (90^\circ - (x))\). Therefore, set \(x = 90^\circ - \theta\).

Step 3: Simplify using identities

Substitute \(x\) in the equation: \(\cos (20 + \theta) = \sin (\theta) = \cos(90^\circ - \theta) = \cos (20 + \theta)\). We have \( 90^\circ - \theta = 20^\circ + \theta \).

Step 4: Solve for \(\theta\)

Rearrange and solve the equation: \(90^\circ = 20^\circ + 2\theta\). Subtract \(20^\circ\) from both sides: \(70^\circ = 2\theta\). Divide by 2 to isolate \(\theta\): \(\theta = 35^\circ\).

Key Concepts

Co-function Identity

In trigonometry, the co-function identity is a valuable tool for relating sine and cosine functions with complementary angles.

The identity states that:

• For any angle \(x\), \( \cos(90^\circ - x) = \sin x \) and \( \sin(90^\circ - x) = \cos x \).

This means that the cosine of an angle is equal to the sine of its complement, and vice versa. Complementary angles add up to 90 degrees. By applying the co-function identity, we can easily convert between sine and cosine functions.

In our exercise, this identity allows us to equate \( \sin \theta \) to \( \cos(20^\circ + \theta) \), by considering the relationship between complementary angles.

Angle Simplification

Angle simplification is a common task in trigonometry, where we simplify expressions to find identical or easier forms.

In the context of the trigonometric identities, simplifying angles often involves using the concept of complementary angles or specific known values. For this exercise, we leverage the equation \( 90^\circ - \theta = 20^\circ + \theta \) to simplify the expression and solve for \( \theta \).

This simplification guides us to equate and eventually solve for the angle \( \theta \) effectively, especially when it involves linear terms.

Equation Solving

Solving equations is a vital skill in mathematics, including when dealing with trigonometric identities.

In the given problem, we are tasked with finding the value of \( \theta \) that satisfies \( \sin \theta = \cos (20^\circ + \theta) \).

• The first step is to set up the equation based on the given identity and known trigonometric properties.
• Next, rearrange the equation to isolate the terms that involve \( \theta \).

After using the co-function identity, solving the equation simplifies to finding the value \( \theta \) that meets the set condition, leading to a straightforward linear equation.

We rearrange and solve \( 90^\circ = 20^\circ + 2\theta \) to find that \( \theta = 35^\circ \).

Sine and Cosine Functions

The sine and cosine functions are fundamental to trigonometry and appear frequently in angle and side relationships in a right triangle.

These functions:

• For angle \(x\), \(\sin x\) represents the ratio of the opposite side to the hypotenuse.
• \(\cos x\) is the ratio of the adjacent side to the hypotenuse.

In the unit circle, these functions can be visualized as the \(y\)-coordinate (sine) and \(x\)-coordinate (cosine) of a point on the circle.

Understanding how these functions relate through identities such as the co-function identity provides powerful methods to solve trigonometric equations, like the one in this exercise. Recognizing how they complement each other aids in simplifying and solving problems efficiently.