In this lesson, we will examine the trigonometric ratios of angles in the four quadrants

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## Quadrant 1 (0˚ < *θ* < 90˚)

## Quadrant II (90˚ < *θ* < 180˚)

## Quadrant III (180˚ < *θ* < 270˚)

## Quadrant IV (270˚ < *θ* < 360˚)

## Videos

Unit Circle, Reference Angle and Signs of Trig Functions in 4 Quadrants.
Finding Trig Functions Given A Point(x, y) in Different Quadrants

You can use the Mathway widget below to practice Trigonometry or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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More Lessons on Trigonometry

Take note of the signs of the trigonometric ratios in the following examples.

In the following diagram, * θ* is in the first quadrant.

Sine, cosine and tangent are all positive.

In the following diagram, *θ* is in the second quadrant.

The reference angle, α = 180˚ – * θ*

Sine is positive whereas cosine and tangent are negative.

In the following diagram, * θ* is in the third quadrant.

The reference angle, α = * θ* – 180˚

Tangent is positive whereas sine and cosine are negative.

In the following diagram, * θ* is in the fourth quadrant.

The reference angle, α = 360˚– * θ*

Cosine is positive whereas sine and tangent are negative.

We can use a mnemonic like **CAST ** or** A**ll** S**tudents **T**ake** C**alculus to remember the signs in the 4 quadrants .

The trigonometric ratios for 0˚, 90˚, 180˚, 270˚ and 360˚ are shown below:

* Example*

Determine the sign of each of the following values.

a) cos 121˚

b) tan 220˚

* Solution: *

a) cos 121˚ is in quadrant II (90˚ *<* 121˚* < * 180˚)

In quadrant II, only sine is positive, so cos121˚ is negative

b) tan 220˚ is in quadrant III (180˚ *<* 220˚* < * 270˚)

In quadrant III, tangent is positive, so tan 220˚ is positive

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