Problem 30 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=-\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions are: \(\sin(t) = \frac{\sqrt{2}}{2}\), \(\cos(t) = -\frac{\sqrt{2}}{2}\), \(\tan(t) = -1\), \(\csc(t) = \sqrt{2}\), \(\sec(t) = -\sqrt{2}\), \(\cot(t) = -1\).

Step 1: Locate the Angle In Unit Circle

The angle \(t = -\frac{3 \pi}{4}\) is the negative equivalent of \(\frac{3 \pi}{4}\) but measured in the clockwise direction. It lands in the second quadrant where sine is positive, cosine and tangent are negative.

Step 2: Find the Values of Sin, Cos and Tan

In the unit circle, the coordinates at \(t = -\frac{3 \pi}{4}\) or \(\frac{\pi}{4}\) in the second quadrant are \(-\frac{\sqrt{2}}{2}\) and \(\frac{\sqrt{2}}{2}\). Thus, the values of sine, cosine and tangent at this angle will be \(\sin(t) = \frac{\sqrt{2}}{2}\), \(\cos(t) = -\frac{\sqrt{2}}{2}\) and \(\tan(t) = -1\). As tangent is the ratio of sine to cosine, the negative signs for cosine and sine cancel.

Step 3: Find the Values of Cosec, Sec, and Cot

The values of the other three trigonometric functions can be found by taking the reciprocals of sine, cosine and tangent. Remember, we cannot take the reciprocal of zero. Thus, the values are \(\csc(t) = \frac{1}{\sin(t)} = \frac{2}{\sqrt{2}} = \sqrt{2}\), \(\sec(t) = \frac{1}{\cos(t)} = -\frac{2}{\sqrt{2}} = -\sqrt{2}\) and \(\cot(t) = \frac{1}{\tan(t)} = -1\).

Key Concepts

Unit Circle

The unit circle is a crucial concept when dealing with trigonometric functions. It is a circle with a radius of exactly 1, centered at the origin of a coordinate plane. The circle helps visualize how sine, cosine, and tangent come to be.
In the unit circle:

• The angle is measured from the positive x-axis, moving counterclockwise.
• The coordinates of any point on the unit circle are expressed as \( ( \cos(\theta), \sin(\theta) ) \).
• The value of tangent can be found by dividing the sine value by the cosine value, i.e., \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

When angles are negative, they are measured clockwise. For example, the angle \( t = -\frac{3\pi}{4} \) is found by moving \( \frac{3\pi}{4} \) in a clockwise direction. This places the angle in the second quadrant of the unit circle, where sine is positive while cosine and tangent are negative.

Sine Cosine Tangent

Sine, cosine, and tangent are the basic trigonometric functions that describe the relationships of angles in a right triangle. Each has a special relationship with the unit circle:
When finding \( \sin(t), \cos(t), \tan(t) \) for an angle \( t \) on the unit circle:

• \( \sin(t) \) represents the y-coordinate of the angle's corresponding point on the unit circle.
• \( \cos(t) \) represents the x-coordinate of the angle's corresponding point on the unit circle.
• \( \tan(t) \) is calculated as the ratio of \( \sin(t) \) to \( \cos(t) \), or \( \tan(t) = \frac{\sin(t)}{\cos(t)} \).The angle \( t = -\frac{3\pi}{4} \) in the second quadrant yields \( \sin(t) = \frac{\sqrt{2}}{2} \), \( \cos(t) = -\frac{\sqrt{2}}{2} \), and \( \tan(t) = -1 \).

Negative signs indicate direction on the cartesian plane, with sine positive pointing upwards, and cosine and tangent negative pointing to the left and downward.

Reciprocal Identities

Reciprocal identities relate to the three remaining trigonometric functions: cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). They are reciprocals of the primary functions:

• \( \csc(t) = \frac{1}{\sin(t)} \)
• \( \sec(t) = \frac{1}{\cos(t)} \)
• \( \cot(t) = \frac{1}{\tan(t)} \)

When \( \sin(t), \cos(t), \) and \( \tan(t) \) have nonzero values, their reciprocals are straightforward to calculate. For the angle \( t = -\frac{3\pi}{4} \):

• \( \csc(t) = \sqrt{2} \)
• \( \sec(t) = -\sqrt{2} \)
• \( \cot(t) = -1 \)

These identities are invaluable when tackling more complex trigonometric equations, allowing for the switching out of sine, cosine, and tangent modules with their reciprocals.