Problem 61 - Trigonometry

Question

Solve each equation for \(\theta\) if \(0^{\circ} \leq \theta<360^{\circ}\). \(\sin \theta+\cos \theta=1\)

Step-by-Step Solution

Answer

\(\theta = 90^{\circ}\) is the solution.

Step 1: Understand the equation

The given equation is \( \sin \theta + \cos \theta = 1 \). We know that in trigonometry, sine and cosine values are between -1 and 1, and we are trying to find the angle \( \theta \) that satisfies this equation within the range.

Step 2: Square both sides

Square both sides of the equation to eliminate the square root complexities: \( (\sin \theta + \cos \theta)^2 = 1^2 \). This simplifies to \( \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta = 1 \).

Step 3: Use Pythagorean Identity

According to the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute this into the equation: \( 1 + 2 \sin \theta \cos \theta = 1 \).

Step 4: Simplify the equation

Subtract 1 from both sides: \( 2 \sin \theta \cos \theta = 0 \). Thus, the equation simplifies to finding the root, \( \sin \theta \cos \theta = 0 \).

Step 5: Analyze the product of sine and cosine

The equation \( \sin \theta \cos \theta = 0 \) means either \( \sin \theta = 0 \) or \( \cos \theta = 0 \). Find \( \theta \) that makes these expressions zero within the given range.

Step 6: Solve for \( \sin \theta=0 \)

\( \sin \theta = 0 \) when \( \theta = 0^{\circ}, 180^{\circ} \).

Step 7: Solve for \( \cos \theta=0 \)

\( \cos \theta = 0 \) when \( \theta = 90^{\circ}, 270^{\circ} \).

Step 8: Compare solutions against the original equation

Verify which of these solutions also satisfy the original equation \( \sin \theta + \cos \theta = 1 \). Check each potential solution.

Step 9: Verification

\( \theta = 90^{\circ} \) is a viable solution as it satisfies \( \sin 90^{\circ} + \cos 90^{\circ} = 1 + 0 = 1 \). No other angle from the set \( \{0^{\circ}, 180^{\circ}, 270^{\circ}\} \) will satisfy the original equation.

Key Concepts

Sine and Cosine

In trigonometry, sine and cosine are two of the most fundamental functions. They help us understand and calculate the properties of triangles, especially right-angled triangles. Both functions describe the relationship between the angles and sides of a triangle. For any given angle \( \theta \):

• \( \sin \theta \) is the ratio of the opposite side to the hypotenuse.
• \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse.

These values vary between -1 and 1 due to the unit circle's nature, where the radius (hypotenuse) is constant. The unit circle is a circle with a radius of 1 centered at the origin on the coordinate plane, which makes the calculations of sine and cosine straightforward because they correspond directly to the y and x coordinates of a point on the circle respectively. Understanding these basic properties will allow you to easily work with trigonometric equations like \( \sin \theta + \cos \theta = 1 \). A quick inspection of this equation suggests that there must be a special angle \( \theta \) within the 0° to 360° range that satisfies this relationship.

Angle Solutions

Solving trigonometric equations often involves finding all possible angle solutions that meet the criteria of the given equation. When confronted with the equation \( \sin \theta + \cos \theta = 1 \), we are tasked with identifying the values of \( \theta \) that satisfy it within a complete rotation range of 0° to 360°. This requires a careful analysis of trigonometric functions to see when their sum equals 1. To simplify, the process involves considering when each function, \( \sin \theta \) or \( \cos \theta \), individually reaches values that allow their sum to be equivalent to 1. In this specific problem, determination of significant angles shows that only \( \theta = 90^{\circ} \) satisfies the equation because:

• At \( \theta = 90^{\circ} \), \( \sin 90^{\circ} = 1 \) and \( \cos 90^{\circ} = 0 \)
• This produces \( \sin \theta + \cos \theta = 1 + 0 = 1 \)

This is the only angle in the range which maintains the equality as expected. It's essential to analyze both \( \sin \theta \) and \( \cos \theta \) at strategic angles to confirm their validity as solutions in trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, reflecting the inherent relationship between sine and cosine. It states that for any angle \( \theta \), the sum of the square of \( \sin \theta \) and \( \cos \theta \) equals 1. In mathematical terms: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This identity is derived from the Pythagoras theorem applied to the unit circle.
When analyzing trigonometric equations, this identity is often used to simplify calculations and reduce complex expressions. In our exercise, by squaring the initial equation, the utilization of the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) simplifies the problem solving substantially by eliminating terms, ultimately narrowing down the possible solutions. By applying this identity, we're left with just as simple linear equation \(2 \sin \theta \cos \theta = 0\), further breaking down the problem into manageable parts that reveal possible angle solutions. This showcases the power of the Pythagorean identity in turning seemingly complex trigonometric problems into straightforward calculations.