Problem 31 - Precalculus: A Concise Course

Question

Evaluate (if possible) the six trigonometric functions of the real number. $$ t=\frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

The six trigonometric functions evaluated at \(t = frac{3 \pi}{4}\) are: \(sin(t) = frac{\sqrt{2}}{2}\), \(cos(t) = -frac{\sqrt{2}}{2}\), \(tan(t) = -1\), \(cot(t) = -1\), \(sec(t) = -\sqrt{2}\), \(csc(t) = \sqrt{2}\)

Step 1: Evaluate the sine function

To evaluate the sine function at \(t = frac{3 \pi}{4}\), recall that \(sin(frac{\pi}{4}) = frac{\sqrt{2}}{2}\). So, \(sin(frac{3 \pi}{4}) = sin(\pi - frac{\pi}{4}) = sin(frac{\pi}{4}) = frac{\sqrt{2}}{2}\)

Step 2: Evaluate the cosine function

The cosine function follows a similar pattern. Recall that \(cos(frac{\pi}{4}) = frac{\sqrt{2}}{2}\). But cos function is negative in the second quadrant where the angle \(t = frac{3 \pi}{4}\) lies. So, \(cos(frac{3 \pi}{4}) = - cos(\pi - frac{\pi}{4}) = - cos(frac{\pi}{4}) = -frac{\sqrt{2}}{2}\)

Step 3: Evaluate the tangent function

The tangent function tan(x) is sin(x) / cos(x). So, \(tan(frac{3 \pi}{4}) = sin(frac{3 \pi}{4}) / cos(frac{3 \pi}{4}) = frac{\sqrt{2}}{-\sqrt{2}} = -1\)

Step 4: Evaluate the cotangent function

The cotangent function cot(x) is the reciprocal of tan(x). So, \(cot(frac{3 \pi}{4}) = 1 / tan(frac{3 \pi}{4}) = 1 / -1 = -1\)

Step 5: Evaluate the secant function

The secant function sec(x) is 1/ cos(x). So, \(sec(frac{3 \pi}{4}) = 1 / cos(frac{3 \pi}{4}) = 1 / -frac{\sqrt{2}}{2} = -\sqrt{2}\)

Step 6: Evaluate the cosecant function

The cosecant function csc(x) is 1/ sin(x). So, \(csc(frac{3 \pi}{4}) = 1 / sin(frac{3 \pi}{4}) = 1 / frac{\sqrt{2}}{2} = \sqrt{2}\)

Key Concepts

Sine Function

The sine function is one of the most basic trigonometric functions. It is used to find the ratio of the opposite side to the hypotenuse in a right triangle. If you're dealing with angles in the unit circle, the sine function gives the y-coordinate of the angle's endpoint. For example, at an angle of \( \frac{3\pi}{4} \), the sine value is computed using the angle adjustment property, which is \( \sin(\pi - \theta) = \sin(\theta) \). Here, \( \theta = \frac{\pi}{4} \), and you find that \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \). This is because \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), reflecting the symmetry across the unit circle.

Cosine Function

The cosine function, like sine, is a foundational trigonometric function. It calculates the ratio of the adjacent side to the hypotenuse in a right triangle. With the unit circle perspective, cosine provides the x-coordinate of a point at a given angle. For \( t = \frac{3\pi}{4} \), the cosine function is negative because the angle lies in the second quadrant. Use \( \cos(\pi - \theta) = -\cos(\theta) \) to find \( \cos\left(\frac{3\pi}{4}\right) \). Knowing \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), you apply the formula to get \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). This negative sign indicates the direction along the unit circle's negative x-axis.

Tangent Function

The tangent function is crucial in trigonometry for determining the ratio of the sine to the cosine of an angle. In practical terms, this translates into the ratio of the opposite side to the adjacent side in a right triangle. For \( t = \frac{3\pi}{4} \), evaluate \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Since \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), the result is \( \tan\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1 \). This outcome illustrates the slope of the line formed by the angle.

Cotangent Function

Cotangent is the reciprocal of the tangent function, providing a different perspective on the relationship between sine and cosine. The cotangent is the ratio of the adjacent side to the opposite side in a right triangle. For an angle like \( \frac{3\pi}{4} \), it is computed as \( \cot(x) = \frac{1}{\tan(x)} \). Given that \( \tan\left(\frac{3\pi}{4}\right) = -1 \), \( \cot\left(\frac{3\pi}{4}\right) \) also equals \( -1 \). This suggests a vertical line along the unit circle, emphasizing the inverse relationship with tangent.

Secant Function

The secant function extends the concept of cosine by taking its reciprocal. It focuses on determining the reciprocal of the x-coordinate in the unit circle. For the angle \( t = \frac{3\pi}{4} \), the secant function is \( \sec(x) = \frac{1}{\cos(x)} \). With \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), the computation yields \( \sec\left(\frac{3\pi}{4}\right) = -\sqrt{2} \). This provides insight into how secant extends the cosine's domain and range.

Cosecant Function

Cosecant is the reciprocal of the sine function and serves as its complement. This function helps determine the reciprocal of the y-coordinate in the unit circle perspective. When examining \( t = \frac{3\pi}{4} \), calculate \( \csc(x) = \frac{1}{\sin(x)} \). Since \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \), the result is \( \csc\left(\frac{3\pi}{4}\right) = \sqrt{2} \). It highlights how cosecant operates outside of sine's typical values by providing a broader range.