Problem 11 - Algebra 2

Question

Find the exact value of each expression. \(\sin 135^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 135° = \frac{\sqrt{2}}{2} \).

Step 1: Understand the Quadrant

The angle 135° is in the second quadrant. In this quadrant, sine values are positive since they correspond to the y-component in the unit circle.

Step 2: Use the Reference Angle

In the second quadrant, the reference angle for 135° is found by subtracting it from 180°: \( 180° - 135° = 45° \). This means we can use the sine value of the reference angle 45°.

Step 3: Recall the Sine of 45°

The sine of common angles like 45° is often memorized. The exact value of \( \sin 45° \) is \( \frac{\sqrt{2}}{2} \).

Step 4: Apply the Sine Value to the Given Angle

Since the angle 135° is in the second quadrant and \( \sin \) is positive there, the sine value of 135° is the same as the sine of the reference angle:\( \sin 135° = \sin 45° = \frac{\sqrt{2}}{2} \).

Key Concepts

Sine function

The sine function is one of the fundamental functions in trigonometry. It relates a given angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. In mathematical terms, for an angle \( \theta \), the sine function is expressed as:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

In the context of a unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane, the sine of an angle is the y-coordinate of the corresponding point on the circle. This makes the sine function quite powerful for understanding angles and their relationships because it dynamically changes as the angle moves around the circle.
One crucial aspect of the sine function is its periodic nature, repeating every \( 360^\circ \) or \( 2\pi \) radians. This periodicity makes it a vital tool in both pure mathematics and applied fields like physics and engineering.

Quadrants in unit circle

The unit circle is a crucial tool in trigonometry because it helps to visualize the angles and their sine, cosine, and tangent values. The circle is divided into four quadrants, each indicating a range of angle measurements and the signs of trigonometric functions within those ranges.
Here is how the quadrants are laid out:

• **Quadrant I**: Angles between \( 0^\circ \) and \( 90^\circ \). Here, both sine and cosine values are positive because the points lie in the first quadrant.
• **Quadrant II**: Angles between \( 90^\circ \) and \( 180^\circ \). In this quadrant, sine is positive but cosine is negative. This is because the x-values (cosine) are negative, and y-values (sine) are positive.
• **Quadrant III**: Angles between \( 180^\circ \) and \( 270^\circ \), where both sine and cosine are negative since both x and y coordinates in this quadrant are negative.
• **Quadrant IV**: Angles between \( 270^\circ \) and \( 360^\circ \) or \( 0^\circ \), where sine is negative and cosine is positive, as points start to approach the positive x-axis again.

Understanding these quadrants is fundamental to knowing the signs and behaviors of trigonometric functions like the sine function.

Reference angles

Reference angles are helpful in simplifying calculations within trigonometry by comparing angles to a known acute angle. The reference angle is always the smallest angle that can be formed from the terminal side of a given angle to the x-axis.
To determine a reference angle, take the absolute difference between the given angle and the nearest x-axis angle. Here’s how it works for each quadrant:

• In **Quadrant I**, the reference angle is the angle itself because it is already acute.
• In **Quadrant II**, subtract the angle from \( 180^\circ \). For example, the reference angle for \( 135^\circ \) is \( 45^\circ \).
• In **Quadrant III**, subtract \( 180^\circ \) from the angle. For instance, the reference angle for \( 225^\circ \) is \( 45^\circ \).
• In **Quadrant IV**, subtract the angle from \( 360^\circ \). For example, the reference angle for \( 315^\circ \) is \( 45^\circ \).

Reference angles are powerful tools because they help simplify and unify the process of finding trigonometric functions of any angle, ultimately allowing students to rely on the known values of trigonometric ratios like \( \sin 45^\circ \), which is \( \frac{\sqrt{2}}{2} \).