How to Find Sin 3pi 4 Without A Calculator

Understanding sin(3π/4)

The angle 3π/4 radians is equivalent to 135 degrees. This angle is located in the second quadrant of the unit circle, where the sine function is positive. The reference angle for 3π/4 is π/4 (45 degrees), which has a known sine value of √2/2.

This identity shows that the sine of an angle in the second quadrant is equal to the sine of its reference angle in the first quadrant.

Using Reference Angle

The reference angle concept simplifies calculations for angles in different quadrants. For 3π/4:

1. Identify the quadrant: 3π/4 is between π/2 and π (second quadrant).
2. Find the reference angle: π - 3π/4 = π/4.
3. Use the known sine value for π/4: √2/2.
4. Apply the sign based on the quadrant: Sine is positive in the second quadrant.

Unit Circle Approach

The unit circle is a circle with radius 1 centered at the origin. Any angle θ corresponds to a point (cosθ, sinθ) on the unit circle.

1. Draw the unit circle and mark the angle 3π/4 (135°).
2. Find the coordinates of the point where the terminal side intersects the circle.
3. The y-coordinate of this point is sin(3π/4).
4. Using symmetry and known values, determine the coordinates are (-√2/2, √2/2).

The y-coordinate gives us sin(3π/4) = √2/2.

Trigonometric Identities

Several identities can help find sin(3π/4):

• sin(π - θ) = sinθ (sine of supplementary angles)
• sin(θ) = cos(π/2 - θ) (co-function identity)
• sin(3π/4) = sin(π - π/4) = sin(π/4) = √2/2

These identities show that sin(3π/4) is equal to √2/2, which is approximately 0.7071.

Worked Example

Let's find sin(3π/4) using the reference angle method:

1. Identify the angle: 3π/4 radians (135°).
2. Determine the quadrant: Second quadrant (π/2 < 3π/4 < π).
3. Find the reference angle: π - 3π/4 = π/4.
4. Recall that sin(π/4) = √2/2.
5. Since sine is positive in the second quadrant, sin(3π/4) = √2/2.

This step-by-step approach confirms that sin(3π/4) equals √2/2.