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More Lessons on Trigonometry

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

**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

More Lessons on Trigonometry

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

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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