As we've seen, the cosine and sine ratios for acute angles can be defined in the following ways:

• Using a right triangle. If is an acute angle in a right triangle, then

• Using the unit circle. If is a central angle made by a point on the unit circle in the first quadrant, then

We can't define trigonometric ratios for non-acute angles using right triangles because a right triangle can't have angles larger than However, we can extend these ratios for non-acute angles using unit circles.

To do that, let's consider the unit circle with a central angle in any quadrant, measured in the usual way.

We define the sine and cosine of as follows:

In other words,

• the cosine of is defined to be equal to the -coordinate of and

• the sine of is defined to be equal to the -coordinate of

This way, our trigonometric ratios make sense for any angle, even if this angle is larger than

We can use reference angles to evaluate trigonometric ratios for angles outside the first quadrant.

To demonstrate, let's determine the value of corresponding to the -coordinate of the point on the unit circle, as shown below.

The reference angle for is Let's add this to our diagram.

Let be the image of under a reflection across the -axis. This gives a pair of congruent angles of measure as shown.

According to the definition, is the -coordinate of the point But this is the same as the -coordinate of the point taken with the opposite sign.

The -coordinate of equals the cosine of the reference angle. Therefore, we conclude that

So, we simplified the evaluation of by evaluating the cosine of the corresponding reference angle with an additional sign adjustment for the final result.

We can use a similar argument to show that, since we have

In this case, we do NOT change the sign because the -coordinates of and are equal.

We can calculate a trigonometric ratio of any angle by finding the ratio of the corresponding reference angle and adjusting the sign depending on which quadrant our angle lies in.

To do this, we perform the following steps:

1. Find the reference angle

2. Calculate the value of the ratio for

3. Determine whether the resulting ratio is positive or negative.

We must do the last step because the ratio for will always be nonnegative, but the ratio for may not be.

But how do we determine whether the ratio for is positive or negative? This depends on the trigonometric ratio and which quadrant the angle is in.

To help us remember when various trigonometric ratios are positive, we use the mnemonic "All Students Take Calculus."

• All of the trigonometric ratios are positive in quadrant I

• Sine (and only sine) is positive in quadrant II

• Tangent (and only tangent) is positive in quadrant III

• Cosine (and only cosine) is positive in quadrant IV

We can represent this in a so-called CAST diagram, shown below.

For example, let's consider

Step 1: Since is the nd quadrant, the reference angle is

Step 2: The given ratio is and therefore we're interested in

Step 3: The ratio must be positive because the sine ratio is always positive in the nd quadrant. Therefore,

Let's see another example.