Problem 158
Question
For the following exercises, use a graphing calculator to evaluate. $$ \sin \frac{3 \pi}{4} $$
Step-by-Step Solution
Verified Answer
\( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).
1Step 1: Understand the Problem
We need to evaluate the sine of the angle \( \frac{3\pi}{4} \) using a graphing calculator. This angle is given in radians, which is important to note.
2Step 2: Convert Angle to Degrees (Optional)
Although optional, understanding the angle in degrees could be helpful. \( \frac{3\pi}{4} \) radians is equivalent to \( 135^{\circ} \).
3Step 3: Locate the Angle on the Unit Circle
Identify that \( \frac{3\pi}{4} \) radians, or \( 135^{\circ} \), is located in the second quadrant of the unit circle. In this quadrant, the sine function is positive.
4Step 4: Sine of Special Angles
Recall that \( \sin \frac{3\pi}{4} = \sin (180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).
5Step 5: Verify with Graphing Calculator
Set your calculator to radian mode. Enter \( \sin(\frac{3\pi}{4}) \) and evaluate. You should confirm that the output is \( \frac{\sqrt{2}}{2} \) or approximately 0.707.
Key Concepts
Unit CircleRadiansGraphing CalculatorSine Function
Unit Circle
The unit circle is a crucial concept in trigonometry, and it is a circle with a radius of one. Its center is located at the origin of the coordinate system, at point (0,0). Understanding the unit circle can help you easily discover the values of trigonometric functions like sine and cosine. This tool aids you in visualizing how angles and their corresponding function values relate to each other.
When an angle is drawn in standard position on the unit circle, its vertex is at the center, and its initial side lies along the positive x-axis. The terminal side, depending on the angle measured in radians or degrees, determines the corresponding point on the circle. This point's coordinates are \(\cos(\theta), \sin(\theta)\), where \(\theta\) is the angle. For example, the point at an angle of \(\frac{3\pi}{4}\) radians, which is in the second quadrant, would have coordinates \(\left(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), with the sine value being the y-coordinate.
When an angle is drawn in standard position on the unit circle, its vertex is at the center, and its initial side lies along the positive x-axis. The terminal side, depending on the angle measured in radians or degrees, determines the corresponding point on the circle. This point's coordinates are \(\cos(\theta), \sin(\theta)\), where \(\theta\) is the angle. For example, the point at an angle of \(\frac{3\pi}{4}\) radians, which is in the second quadrant, would have coordinates \(\left(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), with the sine value being the y-coordinate.
Radians
Radians are a way to measure angles, and they are often used in many mathematics contexts instead of degrees. One complete revolution around a circle in radians is equivalent to \(2\pi\). This measure is based on the radius of the circle, making it a natural way to express angles in terms of \(\pi\).
To convert an angle from radians to degrees, you can use the formula: \[\text{degrees} = \text{radians} \times \frac{180}{\pi}\] This conversion is useful when you wish to visualize or interpret angles more intuitively. For instance, the angle \(\frac{3\pi}{4}\) radians converts to \(135^{\circ}\), providing a clearer understanding that it lies in the second quadrant of the unit circle.
To convert an angle from radians to degrees, you can use the formula: \[\text{degrees} = \text{radians} \times \frac{180}{\pi}\] This conversion is useful when you wish to visualize or interpret angles more intuitively. For instance, the angle \(\frac{3\pi}{4}\) radians converts to \(135^{\circ}\), providing a clearer understanding that it lies in the second quadrant of the unit circle.
Graphing Calculator
A graphing calculator is an invaluable tool in exploring and understanding trigonometric functions like sine. These calculators can handle function evaluations in both degrees and radians, making them versatile in problem-solving scenarios.
- Ensure your calculator is set to the correct mode, depending on whether you're working with degrees or radians.
- To evaluate \(\sin\left(\frac{3\pi}{4}\right)\), you would enter this exactly as shown in radian mode.
- The calculator should then provide an output of approximately \(0.707\), which aligns with \(\frac{\sqrt{2}}{2}\).
Sine Function
The sine function is one of the primary trigonometric functions used to explore relationships in triangles and the unit circle. It associates every angle with the y-coordinate of the corresponding point on the unit circle.
Among the properties of the sine function:
Among the properties of the sine function:
- The sine of an angle is positive in the first and second quadrants.
- For special angles, such as \(45^{\circ}\) or \(\frac{\pi}{4}\), the sine value is known to be \(\frac{\sqrt{2}}{2}\).
- Using symmetry: \(\sin(\pi - \theta) = \sin(\theta)\), which helps determine that \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
Other exercises in this chapter
Problem 156
For the following exercises, use a graphing calculator to evaluate. $$ \sin \frac{\pi}{10} $$
View solution Problem 157
For the following exercises, use a graphing calculator to evaluate. $$ \cos \frac{\pi}{10} $$
View solution Problem 159
For the following exercises, use a graphing calculator to evaluate. $$ \cos \frac{3 \pi}{4} $$
View solution Problem 160
For the following exercises, use a graphing calculator to evaluate. $$ \sin 98^{\circ} $$
View solution