In these lessons, 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** 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.
Determine the sign of each of the following values.
a) cos 121°
b) tan 220°
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
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?
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?
If sec θ = -2 and θ is in quadrant III, find all trigonometric function values for θ
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