To find the sine, cosine, and tangent of the angle 5π/4 radians, we first need to identify which quadrant this angle lies in. The angle 5π/4 can be converted to degrees for better understanding:
5π/4 radians = (5 × 180°)/4π = 225°.
This angle is located in the third quadrant, where both sine and cosine values are negative. The reference angle for 5π/4 radians is:
Reference angle = 225° – 180° = 45°.
Now, we can use the known values of sine, cosine, and tangent for the reference angle:
- For 45° (or π/4 radians):
- sin(45°) = √2/2
- cos(45°) = √2/2
- tan(45°) = 1
Since 5π/4 is in the third quadrant, we apply the signs:
- sin(5π/4) = -√2/2
- cos(5π/4) = -√2/2
- tan(5π/4) = 1
In summary, the trigonometric values for 5π/4 radians are:
- sin(5π/4) = -√2/2
- cos(5π/4) = -√2/2
- tan(5π/4) = 1
This means that at an angle of 5π/4 radians, the sine and cosine are both negative, while the tangent is positive, reflecting the properties of angles in the third quadrant.