OML Search

Signs Of Trigonometric Ratios

Related Topics:
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?

We can use a mnemonic like CAST or All Students Take Calculus 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.

Quadrant 1 (0˚ < θ < 90˚)

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

Sine, cosine and tangent are all positive.

trigonometric ratios




Quadrant II (90˚ < θ < 180˚)

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.

signs of trigonometric ratios

Quadrant III (180˚ < θ < 270˚)

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.

trigonometric signs

Quadrant IV (270˚ < θ < 360˚)

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.

trigonometric signs



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

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


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


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.


OML Search


We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


[?] Subscribe To This Site

XML RSS
follow us in feedly
Add to My Yahoo!
Add to My MSN
Subscribe with Bloglines