Determine the value of the trigonometric function for a specific angle

To find the value of the trigonometric function for a specific angle, we first need to identify the standard angle and then use the unit circle properties. Consider the angle $ \theta = \frac{5\pi}{4} $.

The reference angle is $ \frac{\pi}{4} $, and it lies in the third quadrant.

In the third quadrant, both sine and cosine values are negative. Therefore,

$\sin\left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

$\cos\left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

Thus,

$\tan\left( \frac{5\pi}{4} \right) = \frac{\sin\left( \frac{5\pi}{4} \right)}{\cos\left( \frac{5\pi}{4} \right)} = 1 $

To determine the value of the trigonometric function for $ heta = frac{5pi}{4} $, note that this angle is in the third quadrant where sine and cosine are both negative.

The reference angle is $ frac{pi}{4} $.

Using the unit circle,

$sinleft( frac{5pi}{4}
ight) = -sinleft( frac{pi}{4}
ight) = -frac{sqrt{2}}{2} $

$cosleft( frac{5pi}{4}
ight) = -cosleft( frac{pi}{4}
ight) = -frac{sqrt{2}}{2} $

Therefore,

$ anleft( frac{5pi}{4}
ight) = frac{sinleft( frac{5pi}{4}
ight)}{cosleft( frac{5pi}{4}
ight)} = 1 $

Consider $ heta = frac{5pi}{4} $.

The reference angle is $ frac{pi}{4} $, and since it’s in the third quadrant, both sine and cosine are negative:

$sinleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $

$cosleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $

Then,

$ anleft( frac{5pi}{4}
ight) = 1 $

Start Using PopAi Today