Computing Exact Values for Common Angles

Introduction

Key Concepts

The Unit Circle

The unit circle is a circle with a radius of one unit centered at the origin of the coordinate plane. It serves as a pivotal tool in trigonometry for defining the sine and cosine of an angle. Any angle in standard position (vertex at the origin and initial side along the positive x-axis) corresponds to a point \((x, y)\) on the unit circle, where \(x = \cos(\theta)\) and \(y = \sin(\theta)\).

Reference Angles

A reference angle is the acute angle that the terminal side of a given angle makes with the x-axis. Determining the reference angle is crucial for finding exact sine and cosine values, as it simplifies the computation by allowing the use of first-quadrant angle values. The reference angle \(\theta'\) for an angle \(\theta\) can be found using the following:

Special Right Triangles

Special right triangles simplify the process of finding exact trigonometric values. The two commonly used special triangles are:

• 45°-45°-90° Triangle: In this isosceles right triangle, the legs are congruent, and the hypotenuse is \(\sqrt{2}\) times the length of a leg. This yields the exact sine and cosine values for 45°.
• 30°-60°-90° Triangle: This right triangle has side lengths in the ratio \(1 : \sqrt{3} : 2\). It facilitates the calculation of exact sine and cosine values for 30° and 60°.

Exact Values of Sine and Cosine for Common Angles

The most commonly used angles in precalculus are 0°, 30°, 45°, 60°, and 90°, along with their radian equivalents: \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) radians. The exact values of sine and cosine for these angles are derived from the unit circle and special right triangles.

Sine Values:

• \(\sin(0^\circ) = 0\)
• \(\sin(30^\circ) = \frac{1}{2}\)
• \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
• \(\sin(90^\circ) = 1\)

Cosine Values:

• \(\cos(0^\circ) = 1\)
• \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
• \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\cos(90^\circ) = 0\)

Computing Exact Values Using the Unit Circle

To compute the exact sine and cosine values for any common angle, follow these steps:

1. Identify the Angle: Determine the angle for which you need to find the sine and cosine values.
2. Determine the Reference Angle: Use the reference angle to simplify the computation based on the angle's quadrant.
3. Use Special Triangles: Apply the known ratios from special right triangles to find the exact values.
4. Apply Sign Based on Quadrant: Determine the sign of the sine and cosine values based on the angle's quadrant.

Example: Compute \(\sin(150^\circ)\) and \(\cos(150^\circ)\).

Solution:

• The reference angle for 150° is \(180^\circ - 150^\circ = 30^\circ\).
• Since 150° is in the second quadrant, sine is positive and cosine is negative.
• \(\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}\)
• \(\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\)

Radian Measure of Angles

While degrees are commonly used, radian measure is often preferred in higher mathematics. Converting degrees to radians can be done using the relation:

Using this, the exact values of sine and cosine for the standard angles in radians are:

• \(\sin(0) = 0\), \(\cos(0) = 1\)
• \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\), \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
• \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
• \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)
• \(\sin\left(\frac{\pi}{2}\right) = 1\), \(\cos\left(\frac{\pi}{2}\right) = 0\)

Trigonometric Identities for Exact Values

Several trigonometric identities assist in computing exact values for angles beyond the standard ones. Notable identities include:

• Co-function Identities: Relate sine and cosine of complementary angles.
  • \(\sin(90^\circ - \theta) = \cos(\theta)\)
  • \(\cos(90^\circ - \theta) = \sin(\theta)\)
• Even-Odd Identities: Determine the signs of sine and cosine based on the angle's quadrant.
  • \(\sin(-\theta) = -\sin(\theta)\)
  • \(\cos(-\theta) = \cos(\theta)\)
• Sum and Difference Formulas: Facilitate the computation of sine and cosine for the sum or difference of angles.
  • \(\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)\)
  • \(\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)\)

Applying Trigonometric Identities

These identities are instrumental when exact values need to be found for angles not initially present in the standard set. By expressing such angles as sums or differences of standard angles, one can apply the sum and difference formulas to derive the exact sine and cosine values.

Example: Compute \(\sin(75^\circ)\) exactly.

Solution:

• Express 75° as the sum of 45° and 30°: \(75^\circ = 45^\circ + 30^\circ\)
• Apply the sine addition formula:
$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) $$
• Substitute the exact values:
$$ \sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Verifying Exact Values

To ensure accuracy, exact values can be verified using the Pythagorean identity:

Substituting the exact values into this identity should satisfy the equation, confirming the correctness of the computed sine and cosine values.

Example: Verify the exact values for \(\theta = 60^\circ\).

Solution:

• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• Check the Pythagorean identity:
$$ \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1 $$

Applications of Exact Values

Exact trigonometric values are essential in various applications, including solving triangles, modeling periodic phenomena, and analyzing wave functions. They provide precise measurements necessary for engineering, physics, and computer graphics.

Challenges in Computing Exact Values

One of the main challenges is memorizing the exact values for all common angles and understanding how to apply trigonometric identities effectively. Additionally, extending these computations to non-standard angles requires a strong grasp of algebraic manipulation and trigonometric principles.

Strategies for Mastery

Consistent practice in computing exact values, utilizing the unit circle, and applying trigonometric identities are key strategies for mastering this topic. Creating flashcards for exact values and regularly solving related problems can significantly enhance proficiency.

Comparison Table

Summary and Key Takeaways