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.
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 cos121˚ is negative
b) tan 220˚ is in quadrant III (180˚ < 220˚ < 270˚)
In quadrant III, tangent is positive, so tan 220˚ is positive
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