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Which trig functions are positive per quadrant Memory Aid cheat sheet - grade 9-12

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Math Grade 9-12

Which trig functions are positive per quadrant Memory Aid Cheat Sheet

A printable reference covering ASTC, quadrant signs, sine, cosine, tangent, and reciprocal trig functions for grades 9-12.

TopicsTrigonometryFunctionsGraphsMathematicsStudy Guide
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This cheat sheet explains which trigonometric functions are positive in each quadrant of the coordinate plane. Students use it when solving trig equations, graphing trig functions, and evaluating angles beyond the first quadrant. The memory aid All Students Take Calculus gives a quick way to remember the positive trig functions by quadrant.

It helps prevent sign errors when using reference angles and unit circle values.

Quadrant I has all six trig functions positive, Quadrant II has sine and cosecant positive, Quadrant III has tangent and cotangent positive, and Quadrant IV has cosine and secant positive. The core signs come from the coordinates on the unit circle, where cos⁡θ=x\cos\theta = x and sin⁡θ=y\sin\theta = y. Since tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, tangent is positive when sine and cosine have the same sign.

Reciprocal functions always have the same sign as their matching original function.

Key Facts

  • The memory aid ASTC means All Students Take Calculus, matching Quadrants I, II, III, and IV in order.
  • In Quadrant I, all six trig functions are positive: sin⁡θ\sin\theta, cos⁡θ\cos\theta, tan⁡θ\tan\theta, csc⁡θ\csc\theta, sec⁡θ\sec\theta, and cot⁡θ\cot\theta.
  • In Quadrant II, only sine and cosecant are positive: sin⁡θ>0\sin\theta > 0 and csc⁡θ>0\csc\theta > 0.
  • In Quadrant III, only tangent and cotangent are positive: tan⁡θ>0\tan\theta > 0 and cot⁡θ>0\cot\theta > 0.
  • In Quadrant IV, only cosine and secant are positive: cos⁡θ>0\cos\theta > 0 and sec⁡θ>0\sec\theta > 0.
  • On the unit circle, cos⁡θ\cos\theta is the xx-coordinate and sin⁡θ\sin\theta is the yy-coordinate.
  • The tangent sign follows tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, so tangent is positive when sine and cosine have the same sign.
  • Reciprocal functions keep the same sign as their paired functions: csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}, sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}, and cot⁡θ=1tan⁡θ\cot\theta = \frac{1}{\tan\theta}.

Vocabulary

Quadrant
One of the four regions of the coordinate plane formed by the positive and negative xx-axis and yy-axis.
ASTC
A memory aid for which trig functions are positive in Quadrants I through IV: All, Sine, Tangent, Cosine.
Reference angle
The acute angle between the terminal side of an angle and the nearest part of the xx-axis.
Unit circle
A circle with radius 11 centered at the origin, used to define trig values for angles.
Reciprocal function
A trig function made by taking the reciprocal of another trig function, such as sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}.
Terminal side
The ray that forms the final position of an angle after rotating from the positive xx-axis.

Common Mistakes to Avoid

  • Mixing up the quadrant order, which gives the wrong sign pattern. Read ASTC counterclockwise starting in Quadrant I.
  • Forgetting that reciprocal functions keep the same sign as their original functions. Since csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}, cosecant is positive exactly when sine is positive.
  • Assuming a reference angle gives the final sign, which is wrong because reference angles only provide the magnitude. Use the quadrant to decide whether the trig value is positive or negative.
  • Thinking tangent is positive whenever either sine or cosine is positive, which is incorrect. Because tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, tangent is positive only when sine and cosine have the same sign.
  • Using degree measures without locating the quadrant first, which often causes sign errors. For example, 210∘210^\circ is in Quadrant III, so tan⁡210∘\tan 210^\circ is positive.

Practice Questions

  1. 1 Which trig functions are positive for an angle in Quadrant II?
  2. 2 Determine the sign of cos⁡300∘\cos 300^\circ and explain using the quadrant.
  3. 3 Determine whether tan⁡225∘\tan 225^\circ is positive or negative.
  4. 4 Why do sec⁡θ\sec\theta and cos⁡θ\cos\theta always have the same sign?

Understanding Which trig functions are positive per quadrant Memory Aid

The sign pattern comes from direction, not from memorizing isolated facts. Start with an angle in standard position. Its initial side points right along the horizontal axis, and its terminal side ends somewhere around the plane.

A point on that terminal side has a horizontal value and a vertical value. Moving left makes the horizontal value negative. Moving down makes the vertical value negative.

This is the same coordinate reasoning used in graphing. Trigonometry attaches those coordinate directions to ratios, so the signs are a direct result of where the terminal side lies.

Reference angles make this idea useful for actual calculations. A reference angle is the small positive angle between a terminal side and the nearest horizontal axis. It gives the size of a trig value without its final sign.

For example, an angle of one hundred fifty degrees has a reference angle of thirty degrees. The known sine value for thirty degrees gives the magnitude one half.

The location of one hundred fifty degrees then determines whether that result is positive or negative. Keeping magnitude and sign as two separate steps reduces many mistakes.

This method matters when an exercise gives partial information. Suppose cosine has magnitude four fifths and the angle is in the lower left region. The horizontal direction is left, so cosine must be negative.

The vertical direction is down, so sine is negative too. Since tangent comes from one negative quantity divided by another negative quantity, tangent is positive.

Students often make an error by using a calculator result or a memorized special angle before deciding where the angle is located. Find the quadrant first, then apply the sign to the value.

Pay close attention to angles on an axis. They do not belong to any quadrant, so the usual quadrant memory aid does not apply directly. At zero degrees or one hundred eighty degrees, the vertical coordinate is zero.

This makes sine zero and makes cosecant undefined because division by zero is not allowed. At ninety degrees or two hundred seventy degrees, the horizontal coordinate is zero. Cosine is zero there, while secant and tangent are undefined.

These cases appear in unit circle tables, graphs, and trig equations. Mark them clearly instead of forcing them into a sign pattern.