Problem 19 - Contemporary Precalculus

Question

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$3 \pi / 4$$

Step-by-Step Solution

Answer

Short answer: For the angle \(3\pi/4\), the exact values are: \(\sin(3\pi/4) = \frac{1}{\sqrt{2}}\) \(\cos(3\pi/4) = -\frac{1}{\sqrt{2}}\) \(\tan(3\pi/4) = -1\)

Step 1: Finding sine of \(\boldsymbol{3\pi/4}\)

Use the angle subtraction formula: \(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\), with \(\alpha = 180^\circ\) (\(\pi\)) and \(\beta = 45^\circ\) (\(\pi/4\)), which gives us: $$\sin\left(\frac{3\pi}{4}\right) = \sin\left(\pi - \frac{\pi}{4}\right)=\sin(\pi)\cos\left(\frac{\pi}{4}\right)-\cos(\pi)\sin\left(\frac{\pi}{4}\right)$$ We know that \(\sin(\pi)= 0\), \(\cos(\pi)= -1\), and \(\sin(\pi/4)= \cos(\pi/4)= \frac{1}{\sqrt{2}}\). So, $$\sin\left(\frac{3\pi}{4}\right)=0\cdot\frac{1}{\sqrt{2}}-(-1)\cdot\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}$$

Step 2: Finding cosine of \(\boldsymbol{3\pi/4}\)

Use the angle subtraction formula: \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\), with \(\alpha = 180^\circ\) (\(\pi\)) and \(\beta = 45^\circ\) (\(\pi/4\)), which gives us: $$\cos\left(\frac{3\pi}{4}\right) = \cos\left(\pi - \frac{\pi}{4}\right)=\cos(\pi)\cos\left(\frac{\pi}{4}\right)+\sin(\pi)\sin\left(\frac{\pi}{4}\right)$$ We have already determined the values of \(\sin(\pi)\), \(\cos(\pi)\), and \(\sin(\pi/4)\), so $$\cos\left(\frac{3\pi}{4}\right)=-1\cdot\frac{1}{\sqrt{2}}+0\cdot\frac{1}{\sqrt{2}}=-\frac{1}{\sqrt{2}}$$

Step 3: Finding tangent of \(\boldsymbol{3\pi/4}\)

Use the formula: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), so we have: $$\tan\left(\frac{3\pi}{4}\right)=\frac{\sin\left(\frac{3\pi}{4}\right)}{\cos\left(\frac{3\pi}{4}\right)}=\frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}}=-1$$

Step 4: Writing the final answer

The exact values of the sine, cosine, and tangent of \(3\pi/4\) are: $$\sin\left(\frac{3\pi}{4}\right)=\frac{1}{\sqrt{2}}$$ $$\cos\left(\frac{3\pi}{4}\right)=-\frac{1}{\sqrt{2}}$$ $$\tan\left(\frac{3\pi}{4}\right)=-1$$

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that describes the relationship between the angle and side lengths in a right triangle. Specifically, for an angle \( \theta \) in a right triangle, \( \sin(\theta) \) is the ratio of the length of the side opposite the angle to the length of the hypotenuse. It is defined on the unit circle as well, where \( \sin(\theta) \) becomes the y-coordinate of the point on the circle corresponding to the angle \( \theta \).

In our original problem, we consider \( \sin(3\pi/4) \). This angle is located in the second quadrant of the unit circle. To find its value without a calculator, we utilized the angle subtraction formula where \( 3\pi/4 \) is broken down into \( \pi - \pi/4 \). The sine of \( \pi - \theta \) reflects the sine function's property of staying positive in the second quadrant, which leads us to the exact value of \( \sin(3\pi/4) = \frac{1}{\sqrt{2}} \).

Cosine Function

The cosine function is another key trigonometric function, representing the ratio of the adjacent side to the hypotenuse in a right triangle for an angle \( \theta \). On the unit circle, \( \cos(\theta) \) corresponds to the x-coordinate of the point on the circle for an angle \( \theta \).

In solving the given problem for \( \cos(3\pi/4) \), we applied the angle subtraction formula as well. Angles in the second quadrant, like \( 3\pi/4 \), have negative x-values, reflecting in the cosine value. The solution uses the decomposition \( 3\pi/4 = \pi - \pi/4 \) to conveniently connect the angle to well-known values. After applying the formula, it results in \( \cos(3\pi/4) = -\frac{1}{\sqrt{2}} \). The negative sign aligns with cosine being negative in the second quadrant.

Tangent Function

The tangent function links angles to the slope of the line created when drawing the angle from the origin on the unit circle. It is computed as the ratio of the sine and cosine values: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

In our problem involving \( 3\pi/4 \), after determining both the sine and cosine values, calculating the tangent is straightforward. Substituting in the known values gives us \( \tan(3\pi/4) = \frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}} = -1 \). This simple ratio representation of tangent highlights its dependence on both sine and cosine, which are either both positive or both negative in specific quadrants, except in the second quadrant where their signs differ, resulting in a negative tangent value for \( 3\pi/4 \).

Angle Subtraction Formula

The angle subtraction formula is a useful tool in trigonometry for breaking down complex angles into simpler components, allowing for the computation of an angle's sine, cosine, or tangent value. These formulas are:

• \( \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \)
• \( \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \)

In the exercise, we utilized the subtraction formula by expressing \( 3\pi/4 \) as \( \pi - \pi/4 \), thereby simplifying our solution process to use known values, such as \( \sin(\pi/4) \) and \( \cos(\pi/4) \), which have standard trigonometric values. This method not only provides accuracy but also instills a deeper understanding of how trigonometric identities function in practice.

Exact Values in Trigonometry

Exact values are crucial in trigonometry as they represent precise angles without approximation, typically found for standard angles like \( 30^\circ \), \( 45^\circ \), and \( 60^\circ \), corresponding to \( \pi/6 \), \( \pi/4 \), \( \pi/3 \), respectively, in radians.

For \( 3\pi/4 \), which equates to an angle in the second quadrant, exact values play a significant role in determining sine, cosine, and tangent. These values are commonly derived from the unit circle or special right triangles. The exact values for \( \pi/4 \) facilitate easy calculation for angles like \( 3\pi/4 \) by applying transformations. By understanding and utilizing these exact values, you will find solving trigonometry problems more intuitive and error-resistant.