# 5. Signs of the Trigonometric Functions

by M. Bourne

## Angles greater than 90°

### Interactive graph

Don't miss the interactive graph applet below which shows the meaning of the trigonometric ratios for angles > 90^@.

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

x-axis
y-axis

An obtuse angle in standard position.

By Pythagoras, r=sqrt(x^2+y^2. 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 can now 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 1

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,

r=sqrt(x^2+y^2) =sqrt((-2)^2+3^2) =sqrt(4+9)=sqrt13

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

sin theta=y/r=3/sqrt13=0.83205

cos theta=x/r=(-2)/sqrt13=-0.55470

tan theta=y/x=3/-2=-1.5

csc theta=r/y=sqrt13/3=1.2019

sec theta=r/x=sqrt13/-2=-1.80278

cot theta=x/y=(-2)/3=-0.6667

## 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 theta is always going to be positive in the second quadrant.

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

For the tan theta 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 theta is positive, cos theta and tan theta are negative.

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

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

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

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

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

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

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

## Interactive graph

Here's an interactive graph where you can explore trigonometric ratio concepts.

Drag the point P around the curve into all 4 quadrants and observe the sin, cos and tan ratios that result. Notice in particular the ratios which are positive in each quadrant.

Also, note that sin theta is defined as y/r, and cos theta is defined as x/r and tan theta is defined as y/x.

This is where the positive and negative values come from, and it's just an extension of the ratios we learned earlier, when theta was restricted to being less than 90°.

### Examples 2

What is the sign (+ or −) of the following?

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.

a. sin 50^@ is positive.

(first quadrant - all are positive)

b. cos 100^@ is negative

(100^@ is in the second quadrant)

c. tan 200^@ is positive.

(200^@ is in the 3rd quadrant, and tan is positive there).

d. csc 300^@ is negative

(300^@ is in the 4th quadrant and sin is negative there, so it follows that csc will also be negative).

### Exercises

1. What is the sign (+ or −) of the following?

a) sin(100^@)

b) sec(-15^@)

c) cos(188^@)

a. sin(100^@) is positive.

(100^@ is in the 2nd quadrant, and sin is positive there)

b. sec(-15^@) is positive.

(-15^@ is in the fourth quadrant, since negative angles are measured clockwise. Since cos is positive in the 4th quadrant, so sec will be also.)

c. cos(188^@) is negative.

(188^@ is in the 3rd quadrant, and cos is negative there).

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

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

x = -3 and y = -4.

r=sqrt((-3)^2+(-4)^2) =sqrt(9+16) =sqrt25=5

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

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