5. Signs of the Trigonometric Functions

by M. Bourne

Angles greater than 90°

Flash Interactive

Don't miss the interactive Flash applet which shows the meaning of the trigonometric ratios for angles > 90°.

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Flash trigonometry applet

We define the trigonometric functions for angles greater than 90° in the following way:

math expression

By Pythagoras, r = . Then the ratios are:

sin θ = y/r

cos θ = x/r

tan θ = y/x

csc θ = r/y

sec θ = r/x

cot θ = x/y

How is this different to the definitions we already met in section 2, Sine, Cosine, Tangent and the Reciprocal Ratios? The only difference is that now x or y (or both) can be negative because our angle now can be in any quadrant. It follows that the trigonometric ratios can turn out to be negative or positive. In the earlier section, the angles involved were always less than 90° so all 6 ratios were positive.

Notice that r is always positive.

Example

Let's see how the trigonometric ratios are defined using a particular example. Let our angle θ be defined by the point (-2,3) in the following way:

By Pythagoras,

For this example, we define the trigonometric ratios for θ in the following way:

math expression

The Four Quadrants - Positive or Negative?

Observe for the example above, that our angle was in the second quadrant. Also notice that in the second quadrant, the y-value is positive. Since r is always positive, then y/r will always be positive in quadrant II. So we conclude sin θ is always going to be positive in the second quadrant.

Also observe for (in the cos θ case) , that x was negative. In the second quadrant, x is always negative. So cos θ will always be negative there, too.

For the tan θ case, y is positive and x is negative, so y/x will always be negative.

Considering the other quadrants, we see a pattern.

In Quadrant II, sin θ is positive, cos θ and tan θ are negative.

In Quadrant III, tan θ is positive (both x and y are negative, so y/x is positive), sin θ and cos θ are negative.

In Quadrant IV, cos θ is positive, sin θ and tan θ are negative.

Of course the reciprocal ratios, csc θ, sec θ and cot θ follow the same pattern:

In Quadrant II, csc θ is positive, sec θ and cot θ are negative.

In Quadrant III, cot θ is positive, csc θ and sec θ are negative.

In Quadrant IV, sec θ is positive, csc θ and cot θ are negative.

We don't need to remember the reciprocal ones off by heart, but it is recommended that you remember where sin θ, cos θ and tan θ are positive.

We use this diagram to remember what ratios are positive in each quadrant. We can remember it using:

All Stations To Central.

It means: In the first quadrant (I), all ratios are positive.

In the second quadrant (II), sine (and cosec) are positive.

In the third quadrant (III), tan (and cotan) are positive.

In the fourth quadrant (IV), cos (and sec) are positive.

These just follow from the sign (+ or -) of x or y for each quadrant, as we saw above.

These signs are important when we are finding an angle from a given ratio.

Flash Interactive

Here is a Flash movie to play with. Drag the glowing ball and observe the sin, cos and tan ratios that result. Notice in particular the ratios which are positive in each quadrant. Also, note that sine and cosine are just the ratios y/r, x/r, etc.

Loading Flash movie...

Examples

What is the sign (+ or - ?) of:

a. sin 50°

b. cos 100°

c. tan 200°

d. csc 300°

Do these without calculator so that you have a better idea what is going on.

Answers: a. Positive (first quadrant - all are positive)

b. Negative (100 degrees is in the second quadrant)

c. Positive (200 degrees is in the 3rd quadrant, and tan is positive there.

d. Negative (300 degrees is in the 4th qudrant and sin is negative there, so it follows that csc will also be negative.

Exercises

1. What is the sign (+ or -) of

a) sin(100°)

b) sec(-15°)

c) cos(188°)

Answers: a. Positive (100 degrees is in the 2nd quadrant - sin is positive)

b. Positive (-15 degrees is in the fourth quadrant, since negative angles are measured clockwise. Cos is positive in the 4th quadrant, so sec will be also.)

c. Negative (188 degrees is in the 3rd quadrant, and cos is negative there.

2. Find the trigonometric ratios of the angle with terminal side at (-3,-4).

Answer: The question means "find all of the six ratios: sin, cos, tan, csc, sec, cot for this example".

x = -3 and y = -4.

So

sin θ = y/r = -4/5

cos θ = x/r = -3/5

tan θ = y/x = -4/-3 = 4/3

And for the reciprocal ratios:

csc θ = r/y = -5/4

sec θ = r/x = -5/3

cot θ = x/y = 3/4