Problem 1 - Algebra 2 and Trigonometry

Question

Show that if arcsin \(x=-\frac{1}{2},\) then the measure of the reference angle for \(x\) is \(30^{\circ} .\)

Step-by-Step Solution

Answer

If \( \arcsin(x) = -\frac{1}{2}\), the reference angle is \( 30^{\circ} \).

Step 1: Understanding arcsin

The function \( \arcsin(x) \) gives the angle whose sine is \( x \). So, when \( \arcsin(x) = -\frac{1}{2} \), you are looking for an angle whose sine value is \(-\frac{1}{2}\). The arc sine function returns an angle \( y \) such that \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \).

Step 2: Finding the sine of common angles

Common angle values should be considered to determine where the sine equals \(-\frac{1}{2}\). Know that \( \sin(30^{\circ}) = \frac{1}{2} \). Thus, \( \sin(-30^{\circ}) = -\frac{1}{2} \) since sine is an odd function \( \sin(-x) = -\sin(x) \).

Step 3: Identifying the principal value

Given that \( \sin(-30^{\circ}) = -\frac{1}{2}\), \( -30^{\circ} \) or \( -\frac{\pi}{6} \) is the principal value returned by \( \arcsin(x) \) for \( x = -\frac{1}{2} \), as it lies within the range \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \).

Step 4: Calculating the reference angle

The reference angle is the absolute measure in radians or degrees of \( y \) from the x-axis. As we have \( y = -30^{\circ} \), the reference angle is \( |-30^{\circ}| = 30^{\circ} \). Thus, the reference angle is \( 30^{\circ} \).

Key Concepts

Arcsin Function

The arcsin function, otherwise known as the inverse sine function, is a fascinating part of trigonometry. It's used to determine the angle whose sine is a specific value. For instance, if the sine of a certain angle is \(-\frac{1}{2}\), the arcsin function can help us find that angle.
When we write \( \arcsin(x) \,=\, -\frac{1}{2} \), what we're seeking is an angle in the range of \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) whose sine is exactly \(-\frac{1}{2}\). The arcsin function is crucial because it gives us this specific angle within this limited range, ensuring uniqueness and consistency.

• It helps identify the specific angle that corresponds to a given sine value.
• Its range is limited to the first and fourth quadrants of the unit circle, which are from \(-90^{\circ}\) to \(90^{\circ}\) (or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)).

To solve the problem, we notice that if \( \arcsin(x) \) returns \(-30^{\circ}\), that's because \( \sin(-30^{\circ}) = -\frac{1}{2} \). Understanding this relationship is a key part of using the arcsin function effectively.

Sine Function

The sine function is one of the primary trigonometric functions. It relates a given angle with the ratio of the length of the opposite side to the hypotenuse in a right triangle.
For common angles such as \(30^{\circ},\, 45^{\circ},\) and \(60^{\circ}\), the sine values are memorized due to their frequent occurrence in mathematics.

• For example, \( \sin(30^{\circ}) \,=\, \frac{1}{2} \), which is a fundamental trigonometric value.
• The sine function is periodic and it repeats its values every \(360^{\circ}\) or \(2\pi\) radians.
• Being an odd function, it follows that \( \sin(-x) = -\sin(x) \), which is crucial when considering negative angles, such as \( -30^{\circ} \).

In our exercise, finding that \( \sin(-30^{\circ}) = -\frac{1}{2} \) utilizes this property of odd functions, and helps us identify the angle related to \(-\frac{1}{2}\). This process shows the symmetry and periodicity of the sine function.

Trigonometric Values

Trigonometric values are essentially the sine, cosine, and tangent of standard angles. These values form the backbone of many trigonometric calculations, especially when working with triangles and circles. Knowing these values can simplify solving problems related to angles and their sine in radians or degrees.

• Reference angles are often used to simplify the computation of trigonometric functions for angles outside the first quadrant.
• For the sine function, important reference angles are \(30^{\circ},\) \(45^{\circ},\) and \(60^{\circ}\).
• These angles have recognizable and memorable sine values, such as \( \sin(30^{\circ}) = \frac{1}{2} \).

When calculating the reference angle, remember that it's simply the smallest angle formed with the x-axis, regardless of the initial angle's quadrant. So, for \( -30^{\circ} \), the reference angle is \( 30^{\circ} \), which shows this angle's relationship to the primary angle quadrant-wise. This understanding assists in better establishing the connection between the angles and their trigonometric values.