Trigonometric Functions of Any Angle

The CAST rule you learned in the previous lesson suggests that for every ratio you evaluate using an inverse trigonometric function, there are at least two angles between 0° and 360° that represent that ratio. For example, if we’re given cos θ = 0.3345, using cos-1(0.3345) = θ we get 70.6°. But now we know that cosine gives a positive ratio in the 4th quadrant too. To find the other angle in the 4th quadrant, we take 360° – 70.6° = 289.4°.

Now, let’s try doing this for a negative ratio, for example cos θ = −0.2321. You could evaluate this on your calculator right away to get θ = 103°, but then you’d have to remember that cosine is also negative in the 3rd quadrant. So you’d first have to subtract 180° from 103° to get the reference angle, which is 77°, then add this to 180° to find the second angle that’s in the third quadrant.

Another approach which is used in all the examples performed in the video below is first take the absolute value of the negative ratio (|−0.2321| = 0.2321). Taking the absolute value will ensure you get a positive acute reference angle. Then recall that cosine is negative in the 2nd and 3rd quadrant, so you subtract your acute reference angle from 180° and add to 180°, respectively.

The video below shows this process for tangent.

When dealing with reciprocal trigonometric functions, you’ll have to use the non-reciprocal version of that function. This involves converting the ratio by taking its reciprocal. An example involving secant is shown below:

A few more examples with reciprocal functions are demonstrated underneath. Notice the number of ways the same questions are being asked.

The last video in this section shows how to perform these calculations using radians rather than degrees. The process is the exact same, just make sure your calculator is in radian mode.

  • Note that in some pure trigonometry courses, special angles and the unit circle are used to evaluate problems similar to these. While special angles are introduced in this course (next lesson), they are not used to evaluate such problems given their complexity.