In these lessons, we will examine the trigonometric ratios of angles in the four quadrants.

**Related Pages**

Trigonometric Graphs

Trigonometric Functions

Lessons On Trigonometry

**How to remember the signs of the trigonometric functions for 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 following figure shows the signs of the trigonometric functions for the four quadrants. Scroll down the page for more examples and solutions.

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 cos 121° is negative.

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

In quadrant III, tangent is positive, so tan 220° is positive.

**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

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