Problem 16 - Algebra 2

Question

Find the value of each expression. \(\cos \theta,\) if \(\sin \theta=\frac{1}{2} ; 0^{\circ} \leq \theta<90^{\circ}\)

Step-by-Step Solution

Answer

The value of \( \cos \theta \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Use the Pythagorean Identity

The Pythagorean identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \). Given \( \sin \theta = \frac{1}{2} \), we can substitute this into the equation to solve for \( \cos^2 \theta \).

Step 2: Substitute Into the Identity

Substitute the given value \( \sin \theta = \frac{1}{2} \) into the Pythagorean identity:\[\left(\frac{1}{2}\right)^2 + \cos^2 \theta = 1\]This simplifies to:\[\frac{1}{4} + \cos^2 \theta = 1\]

Step 3: Rearrange the Equation

Subtract \( \frac{1}{4} \) from both sides of the equation:\[\\cos^2 \theta = 1 - \frac{1}{4}\]Thus:\[\cos^2 \theta = \frac{3}{4}\]

Step 4: Solve for \( \cos \theta \)

Take the square root of both sides to solve for \( \cos \theta \):\[\cos \theta = \pm \sqrt{\frac{3}{4}}\]This simplifies to:\[\cos \theta = \pm \frac{\sqrt{3}}{2}\]

Step 5: Determine the Sign of \( \cos \theta \)

Since \( 0^{\circ} \leq \theta < 90^{\circ} \), \( \theta \) is in the first quadrant, where both sine and cosine are positive. Thus, we choose the positive value:\[\cos \theta = \frac{\sqrt{3}}{2}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable within their domains. They provide us tools to simplify complex expressions and solve trigonometric equations.

One fundamental trigonometric identity is the Pythagorean identity, which states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity derives from the geometry of a right triangle and the definition of sine and cosine. In solving problems, it can be used to find one trigonometric function given another, as long as the angle is known to lie within a specific range.

In the case of the exercise given, knowing \( \sin \theta = \frac{1}{2} \) allowed us to find \( \cos \theta \) using this identity. By substituting the sine value into the identity, we gain a straightforward path to solve for the cosine of the angle.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is a powerful tool in understanding trigonometric functions and their relationships. In the unit circle, any point \( (x, y) \) on the circle can represent the cosine and sine of an angle \( \theta \), respectively, from the positive x-axis. That is to say:

• \( x = \cos \theta \)
• \( y = \sin \theta \)

The angle \( \theta \) is measured from the positive x-axis to the line segment joining the origin to the point on the circle, and it is measured in a counter-clockwise direction.

For angles between \( 0^{\circ} \) and \( 90^{\circ} \), known as the first quadrant, both sine and cosine values are positive. This property is crucial in determining the correct sign of trigonometric expressions, such as \( \cos \theta \), when applying identities or solving equations. In the problem above, since \( \theta \) was within this range, \( \cos \theta \) was positive.

Pythagorean Identity

The Pythagorean identity is a special kind of trigonometric identity that relates the square of sine and cosine of an angle \( \theta \) to 1:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

Derived from the Pythagorean Theorem, this identity represents a relationship bound by the unit circle where the hypotenuse of a right triangle is always 1.

In our problem, we used the Pythagorean identity to solve for \( \cos \theta \), knowing that \( \sin \theta = \frac{1}{2} \). By plugging in the given value of sine into the identity and performing algebraic manipulations, we find that \( \cos^2 \theta = \frac{3}{4} \). Solving for \( \cos \theta \) requires taking the square root. Since \( \theta \) is in the first quadrant, \( \cos \theta \) is positive, leading us to the final answer of \( \cos \theta = \frac{\sqrt{3}}{2} \).

The Pythagorean identity is often the backbone of trigonometric problem-solving, enabling connections between different trigonometric ratios.