Finding Trig Ratios for Larger Angles

To refresh:

How do you find trig ratios for angles more than ninety degrees?

To find the values of trigonometric ratios when the angles are greater than 90°, follow these steps:

1. In the x,y-plane, draw the terminal side of the angle.

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2. Draw a line from this terminal line to the x-axis, perpendicular to the x-axis.

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3. Within the right triangle formed by the axis, the terminus line, and the perpendicular, label the base angle (at the origin) with its size within that quadrant of the plane. For instance, an angle of 120° becomes a triangle in the second quadrant, with a base-angle measure of 180° − 120° = 60°.
4. Label the axis length and the perpendicular with their values in that quadrant. For instance, for the 60° angle in the second quadrant, the "opposite" value (being the height y) is positive, but the "adjacent" side (along the x-axis) is negative.
5. Use the Pythagorean Theorem to find the value of the hypotenuse, if needed. (This value will always be positive.)
6. Once all three sides are labelled, with their signs, read off the ratios directly from the picture.

With just a little practice, the above process should become pretty easy to do. Let's look at an example.

• Find the six trigonometric ratios for the angle having point (4, −3) on its terminal side.

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Sometimes you'll be given some fragmentary information, from which you are asked to figure out the quadrant for the context.

• Determine the quadrant in which lies the terminal side of the angle θ, given that tan(θ) < 0 and sin(θ) < 0.

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The thought process for the exercise above leads to a rule for remembering the signs on the trig ratios in each of the quadrants.

What are the signs of sine, cosine, and tangent in the four quadrants?

Using the signs of x and y in each of the four quadrants, and using the fact that the hypotenuse r is always positive, we find the following:

• In the first quadrant, all the values (x, y, and r) are positive, so All the trig ratios are positive.
• In the second quadrant, the x-values are negative, so x/r and y/x are negative; only y/r is positive, so only the Sine is positive in QII.
• In the third quadrant, each of x and y is negative, so x/r and y/r are negative; only y/x is positive, so only the Tangent is positive in QIII.
• In the fourth quadrant, the y-values are negative, so y/r and y/x are negative; only x/r is positive, so only the Cosine is positive in QIV.

You're probably wondering why I capitalized the trig ratios and the word "All" in the preceding paragraph. I did that to explain this picture:

The letters in the quadrants stand for the initials of the trig ratios which are positive in that quadrant.

What mnemonics can help you remember where each of sine, cosine, and tangent is positive?

Some people remember the letters indicating positivity by using the word "ACTS", but that's the reverse of normal (anti-clockwise) trigonometric order. Others remember the letters with the word "CAST", which is the normal rotational order but doesn't start in the usual (first-quadrant) starting place.

To start in the usual spot and rotate in the usual direction, still others use the mnemonic "All Students Take Calculus" (which is so not true). Use whichever method works best for you.

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• Find the values of the remaining trigometric ratios, given that and θ lies in QIII.