Sine and Cosine of Angles Past 90 Degrees

Sine and cosine as coordinates

Draw a circle of radius 1 with its center at the origin. This is called the unit circle. Start with a radius lying along the positive x-axis, and turn it counterclockwise through an angle $\theta$θ. Its end reaches a point on the circle.

The sine and cosine of $\theta$θ are defined from that point: its x-coordinate is $\cos \theta$cos θ and its y-coordinate is $\sin \theta$sin θ. The point is $\left(\cos \theta ,\sin \theta \right)$(cos θ, sin θ).

For an acute angle this is the same as the triangle definition. Drop a line from the point straight down to the x-axis. That makes a right triangle whose hypotenuse is the radius, 1. The adjacent side is the x-coordinate and the opposite side is the y-coordinate, so $\cos \theta =$cos θ = adjacent ÷ 1 = x and $\sin \theta =$sin θ = opposite ÷ 1 = y.

At 60°, for example, the point is $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$(½, √3/2): cos 60° = ½ and $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$sin 60° = √3/2, the exact values from the half equilateral triangle.

The radius at 60° on the unit circle. The distance across is cos 60° = ½ and the height is $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$sin 60° = √3/2.

Turning past 90°

A right triangle cannot have an angle of 120°. Its right angle and a 120° angle would already add up to 210°, and the angles of a triangle add up to 180°. So the triangle definition has nothing to say about sin 120°. The circle does.

Keep turning the radius past 90°. At 120° the point has crossed to the left of the y-axis, so its x-coordinate is negative. It is still above the x-axis, so its y-coordinate is positive. The definition does not change: cos 120° is the x-coordinate and sin 120° is the y-coordinate, whatever their signs.

The radius can keep turning all the way round. At 90° the point is (0, 1), at 180° it is (−1, 0), at 270° it is (0, −1), and at 360° it is back at (1, 0). So sin 90° = 1 and cos 90° = 0, sin 180° = 0 and cos 180° = −1, and sin 270° = −1 and cos 270° = 0.

The point at 120°. It is left of the center, so its cosine is negative: cos 120° = −0.5. It is above the center, so its sine is positive: sin 120° = 0.87, to 2 decimal places. Drag the point round the circle and watch both coordinates change sign.

The reflection in the y-axis

Compare the points at 60° and 120°. The radius at 120° makes an angle of 180° − 120° = 60° with the negative x-axis. So the two radii are mirror images of each other in the y-axis.

Reflecting a point in the y-axis keeps its height and changes the sign of its x-coordinate: (x, y) becomes (−x, y). The point at 60° is $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$(½, √3/2), so the point at 120° is $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$(−½, √3/2).

That gives the two facts at once: $\sin 120^{\circ }=\frac{\sqrt{3}}{2}$sin 120° = √3/2, exactly the same as sin 60°, and cos 120° = −½, the negative of cos 60°.

The points at 60° and 120° are mirror images in the y-axis. The dashed line joining them is level, so their heights are equal: sin 120° = sin 60°. Their distances across are opposite: cos 120° = −½ and cos 60° = ½.

Any angle and 180° minus it

The same argument works for every angle $\theta$θ. The radius at $180^{\circ }-\theta$180° − θ is the reflection in the y-axis of the radius at $\theta$θ, so the two points have the same height and opposite x-coordinates:

$\sin \left(180^{\circ }-\theta \right)=\sin \theta$sin (180° − θ) = sin θ, and $\cos \left(180^{\circ }-\theta \right)=-\cos \theta$cos (180° − θ) = −cos θ.

For example, sin 150° = sin 30° = ½ and $\cos 150^{\circ }=-\cos 30^{\circ }=-\frac{\sqrt{3}}{2}$cos 150° = −cos 30° = −√3/2. A calculator agrees: sin 150° = 0.5 and cos 150° = −0.8660, to 4 decimal places. As a check, ${\left(\frac{1}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2=\frac{1}{4}+\frac{3}{4}=1$(½)² + (√3/2)² = ¼ + ¾ = 1, which it must be, because the point is 1 away from the center.

The signs in each quadrant

The axes cut the circle into four quarters, called quadrants, numbered counterclockwise from the top right. The sign of each coordinate tells you the sign of the sine and the cosine there.

From 0° to 90°, the first quadrant, the point is above and to the right of the center, so the sine and cosine are both positive. From 90° to 180°, the second quadrant, it is above and to the left, so the sine is positive and the cosine negative. From 180° to 270°, the third quadrant, it is below and to the left, so both are negative. From 270° to 360°, the fourth quadrant, it is below and to the right, so the sine is negative and the cosine positive.

To find sin 210°, first find the angle the radius makes with the x-axis: 210° − 180° = 30°. This is called the reference angle. The size of the sine is sin 30° = ½, and the point is in the third quadrant, below the center, so sin 210° = −½. In the same way $\cos 210^{\circ }=-\frac{\sqrt{3}}{2}$cos 210° = −√3/2.

The sign of the tangent past 90° follows from the same signs, since $\tan \theta =\sin \theta /\cos \theta$tan θ = sin θ / cos θ: in the second quadrant a positive sine divided by a negative cosine gives a negative tangent.

The point at 210°, 30° past the negative x-axis and below it. Both coordinates are negative: cos 210° = −0.87 and sin 210° = −0.5, to 2 decimal places. Drag the point into each quadrant in turn and watch which signs change.

Obtuse angles in a triangle

A triangle can have one obtuse angle, between 90° and 180°, and the circle definition gives its sine and cosine. The sine of an obtuse angle is positive, and equal to the sine of its supplement: sin 130° = sin 50°. The cosine of an obtuse angle is negative.

So the sign of the cosine tells you whether an angle in a triangle is acute or obtuse: positive for acute, negative for obtuse. The sine cannot tell you, because 50° and 130° have the same sine.

The usual mistakes

Making the sine of an obtuse angle negative. The point is above the x-axis between 90° and 180°, so the sine is positive: sin 150° = ½, not −½.

Copying the cosine across. The reflection in the y-axis changes the sign of the x-coordinate, so cos 120° = −½, not ½.

Measuring the angle clockwise, or from the y-axis. The angle is turned counterclockwise from the positive x-axis.

Using a calculator set to radians. It reads sin 150 as the sine of 150 radians, which is −0.7149 to 4 decimal places, not 0.5, so check the calculator is in degrees.

The area of a triangle from two sides and the angle between them

The application below uses the area of a triangle found from two sides and the angle between them. A triangle has sides a and b, and C is the angle between them. Take a as the base. The height is the distance from the far end of b straight down to the line of the base.

When C is acute, that height is the opposite side of a right triangle whose hypotenuse is b, so the height is b sin C. When C is obtuse, the height falls outside the triangle, onto the base extended past C. The right triangle there has the angle 180° − C at C, so the height is b sin (180° − C), which is b sin C again.

Either way, area = ½ × base × height = ½ × a × b sin C, written ½ab sin C. The obtuse case works because an angle and its supplement have the same sine.