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

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More Lessons on Trigonometry

**How to remember the signs of the trigonometric functions for the four quadrants?**

### Quadrant 1 (0˚ < *θ* < 90˚)

### Quadrant II (90˚ < *θ* < 180˚)

### Quadrant III (180˚ < *θ* < 270˚)

### Quadrant IV (270˚ < *θ* < 360˚)

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

Example:

Assume that θ is an angle in standard position whose terminal side contains the point (-5,12). Find the exact values of sin θ, cos θ, and tan θ.**How to Find Trig Function Values Given One Value and the Quadrant?**

Example:

If tan θ = -5/12 and &theta is in quadrant IV, find all trigonometric function values for θ.

**How to find all six trigonometric function values given one value (the secant) and the quadrant?**

Example:

If sec θ = -2 and θ is in quadrant III, find all trigonometric function values for θ

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Related Topics:

More Lessons on Trigonometry

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:

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.

* 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

Example:

Assume that θ is an angle in standard position whose terminal side contains the point (-5,12). Find the exact values of sin θ, cos θ, and tan θ.

Example:

If tan θ = -5/12 and &theta is in quadrant IV, find all trigonometric function values for θ.

Example:

If sec θ = -2 and θ is in quadrant III, find all trigonometric function values for θ

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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