Calculus Tutorial

Intro

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

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 (denoted ), it is equal to the side opposite of the angle divided by the hypotenuse of the triangle. (E.g., in the triangle above, since is the side opposite to the angle and is the hypotenuse of the triangle.) Further, when taking the cosine of an angle (denoted ), it is equal to the side adjacent to divided by the hypotenuse. (E.g., since is on the side adjacent to and is the hypotenuse.) Finally, the tangent of an angle (denoted ) is defined to be the side opposite of the angle divided by the side adjacent to the angle . (E.g. since is the opposite side to and is the adjacent side to .)

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 -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 be an angle such that and . What is ?

Solution

Since is negative and is positive, our angle must lie in quadrant IV. Further, since cosine is adjacent over hypotenuse and , this means that the adjacent side (side in the image) has length and the hypotenuse (side in the image) has length .

By the Pythagorean Theorem, so plugging in our values, we have:

Therefore, we know the length of all four sides and we can find . By TOA, we know that tangent of is the opposite side over the adjacent side . Further, we know that we are working in the fourth quadrant, so tangent is negative. Putting this together, we see that .