Question

Use your understanding of the unit circle and trigonometric functions to find the values requested.

a. sin (− ???? / 3)
b. tan (5???? / 4)

Answer 1

a) For this case we can use the fact that
`sin (\pi/3) = (√(3))/(2)`

And for this case since we ar einterested on
`-(\pi)/(3)` and we know that the if we are below the y axis the sine would be negative then:

`sin (-\pi/3) = -(√(3))/(2)`

b) From definition we can use the fact that
`tan x= (sin x)/(cos x)` and we got this:

`tan (5\pi/4) = (sin(5\pi/4))/(cos(5\pi/4))`

We can use the notabl angle
`\pi/4` and we know that :

`sin (\pi/4) = cos(\pi/4) = (√(2))/(2)`

Then we know that
`5\pi/4` correspond to 225 degrees and that correspond to the III quadrant, and we know that the sine and cosine are negative on this quadrant. So then we have this:

`tan (5\pi/4) = (sin(5\pi/4))/(cos(5\pi/4))= \frac{\frac{sqrt{2}}{2}}{(√(2))/(2)} = 1`

Explanation:

For this case we can use the notable angls given on the picture attached.

Part a

For this case we can use the fact that
`sin (\pi/3) = (√(3))/(2)`

And for this case since we ar einterested on
`-(\pi)/(3)` and we know that the if we are below the y axis the sine would be negative then:

`sin (-\pi/3) = -(√(3))/(2)`

Part b

From definition we can use the fact that
`tan x= (sin x)/(cos x)` and we got this:

`tan (5\pi/4) = (sin(5\pi/4))/(cos(5\pi/4))`

We can use the notabl angle
`\pi/4` and we know that :

`sin (\pi/4) = cos(\pi/4) = (√(2))/(2)`

Then we know that
`5\pi/4` correspond to 225 degrees and that correspond to the III quadrant, and we know that the sine and cosine are negative on this quadrant. So then we have this:

`tan (5\pi/4) = (sin(5\pi/4))/(cos(5\pi/4))= ((√(2))/(2))/((√(2))/(2)) = 1`