Step by step
The question
Question: find the exact value of sin 420°.
1 Bring it into 0° to 360°
A full turn is 360°.
Taking away a full turn lands at the same place: 60°.
Same place on the circle, same point.
So sin 420° = sin 60°.
2 Find the quadrant
We turn 60° from the positive x-axis, against the clock.
The ray meets the circle at P.
60° is between 0° and 90°.
So it is in quadrant I, the top right.
3 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 360° − 60°, which is 60°.
Careful!
Always measure to the x-axis, never to the y-axis.
The y-axis gives 30°: the wrong triangle.
4 Find the sign
In quadrant I, P is right of the y-axis: x is positive.
P is above the x-axis: y is positive.
Sin is the up number y.
In quadrant I, y is positive.
So sin 60° = sin 60°: the same size as for 60°, and plus.
5 Use the reference angle's value
From the unit circle: sin 60° = √3/2.
So sin 420° = √3/2.
6 Check
Let's check with the calculator: sin 420° is about 0.8660.
And √3/2 is about 0.8660.
The same!
So it is right.
Let's look back.
First: bring it into 0° to 360°.
Then: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Last: check.