Trigonometry for Calculus

Calculus Tutorial

Trigonometry for Calculus


Before diving into the world of limits, derivatives, and integrals, it is essential to learn some basic trigonometric facts.

General Triangle

Over the years, math teachers have created several mnemonics to help students remembers trigonometric properties. The first is “SOH CAH TOA” (read: sew, kah, tow-ah). This strange sounding series of acronyms stands for “sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.” This means that when taking the function sine of an angle b (denoted sin(b)), it is equal to the side opposite of the angle b divided by the hypotenuse of the triangle. (E.g., in the triangle above, sin(b)=\frac{B}{H} since B is the side opposite to the angle b and H is the hypotenuse of the triangle.) Further, when taking the cosine of an angle b (denoted cos(b)), it is equal to the side adjacent to b divided by the hypotenuse. (E.g., cos(b)=\frac{A}{H} since A is on the side adjacent to b and H is the hypotenuse.) Finally, the tangent of an angle b (denoted tan(a)) is defined to be the side opposite of the angle b divided by the side adjacent to the angle b. (E.g. tan(b)=\frac{B}{A} since B is the opposite side to b and A is the adjacent side to b.)


To apply trigonometric equations to Calculus, one may need to take into account angles which are outside the bounds of a traditional right triangle. The phrase “All Students Take Calculus” (ASTC) stands for “All, Sine, Tangent, Cosine.” This references which trigonometric functions are positive in the four quadrants of the xy-plane. In the Quadrant I (upper right), all trigonometric functions are positive. In Quadrant II (upper left), only sine of an angle is positive, while the other two functions are negative. Quadrant III (lower left) has positive values for the tangent of an angle and negative values for sine and cosine of an angle. Finally, in Quadrant IV (lower right), only the cosine of an angle is positive while sine and tangent are negative.

Sample Problem

Let a be an angle such that sin(a)<0 and cos(a)=\frac{\sqrt{3}}{2}. What is tan(a)?







Angle in xy plane

Since sin(a) is negative and cos(a) is positive, our angle must lie in quadrant IV. Further, since cosine is adjacent over hypotenuse and cos(a)=\frac{\sqrt{3}}{2}, this means that the adjacent side (side B in the image) has length \sqrt3 and the hypotenuse (side H in the image) has length 2.

By the Pythagorean Theorem, A^2+B^2=H^2, so plugging in our values, we have:



    \[B^2 =4-3\]



Therefore, we know the length of all four sides and we can find tan(a). By TOA, we know that tangent of a is the opposite side A over the adjacent side B. Further, we know that we are working in the fourth quadrant, so tangent is negative. Putting this together, we see that tan(a)=-\frac{\sqrt{3}}{1}=-\sqrt{3}.

About The Author

UCSD PhD Student; Math Tutor
I received my Master's and Bachelor's in Mathematics from Bryn Mawr College in 2014. Currently, I am a PhD student at University of California – San Diego. While there, I have taught Calculus I, II, and III as well as Linear Algebra. I have also tutored high school students in mathematics ranging ...
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