The exact value of $\sin(5\pi/4)$ is $-\frac{\sqrt{2}}{2}$.
How to Find the Exact Value of sin 5pi 4?
Finding the exact value of trigonometric functions like $\sin(5\pi/4)$ involves understanding angles in radians, the unit circle, and reference angles. The sine of an angle on the unit circle corresponds to the y-coordinate of the point where the angle's terminal side intersects the circle.
Understanding Radians and the Unit Circle
An angle measured in radians is a way to express angular displacement. A full circle is $2\pi$ radians, which is equivalent to 360 degrees.
The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) of a Cartesian coordinate system. For any angle $\theta$ measured counter-clockwise from the positive x-axis, the coordinates $(x,y)$ of the point where the angle's terminal side intersects the unit circle represent the cosine and sine of that angle, respectively:
- $x = \cos(\theta)$
- $y = \sin(\theta)$
This means that to find $\sin(5\pi/4)$, we need to determine the y-coordinate of the point on the unit circle corresponding to an angle of $5\pi/4$ radians.
Step-by-Step Calculation for sin 5pi/4
Let's break down the process to find the exact value of $\sin(5\pi/4)$:
Step 1: Locate the Angle on the Unit Circle
- Convert to a more intuitive form: $5\pi/4$ can be thought of as $\pi + \pi/4$.
- Visualize the position:
- $\pi$ radians (or 180 degrees) is along the negative x-axis.
- Adding an additional $\pi/4$ radians (or 45 degrees) from the negative x-axis places the angle in the third quadrant.
- In the third quadrant, both the x-coordinate (cosine) and the y-coordinate (sine) are negative.
Step 2: Determine the Reference Angle
The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. It helps us relate the trigonometric values of any angle to those of an acute angle in the first quadrant.
- For an angle $\theta$ in the third quadrant, the reference angle $\alpha$ is given by $\alpha = \theta - \pi$.
- For $5\pi/4$, the reference angle is:
$\alpha = 5\pi/4 - \pi = 5\pi/4 - 4\pi/4 = \pi/4$.
Step 3: Identify the Sign of Sine in the Quadrant
As established in Step 1, the angle $5\pi/4$ lies in the third quadrant. In the third quadrant, the y-coordinates are negative. Therefore, $\sin(5\pi/4)$ will be negative.
Step 4: Use the Reference Angle to Find the Exact Value
- We know the exact value of $\sin(\pi/4)$. This is a common special angle value derived from a 45-45-90 right triangle.
- $\sin(\pi/4) = \frac{\sqrt{2}}{2}$
- Since $\sin(5\pi/4)$ has the same magnitude as $\sin(\pi/4)$ but is negative in the third quadrant, we combine the sign from Step 3 with the value from this step:
- $\sin(5\pi/4) = -\sin(\pi/4) = -\frac{\sqrt{2}}{2}$
Connecting to Decimal Approximations
The exact value, $-\frac{\sqrt{2}}{2}$, can be approximated as a decimal. If you calculate $\sqrt{2} \approx 1.4142$, then $-\frac{1.4142}{2} \approx -0.7071$. This decimal value (-0.7071) corresponds to the y-coordinate of the point on the unit circle for the angle $5\pi/4$, as described in the unit circle concept. However, for the exact answer, the radical form is preferred.
Summary of Special Angle Values (Example for context)
Understanding common exact values for special angles like $\pi/4$ (45 degrees) is crucial:
Angle (Radians) | Angle (Degrees) | $\sin(\theta)$ | $\cos(\theta)$ |
---|---|---|---|
$\pi/4$ | 45 | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ |
By applying the principles of the unit circle, quadrants, and reference angles, we can precisely determine the trigonometric values for various angles.