Using the Quotient and Pythagorean IdentitiesĪt first, this integral seems impossible! However with a bit of skillful manipulation, this one transforms into an easier problem.įirst let’s re-express the division by cos 2 x as a multiplication by its reciprocal, sec 2 x. Note how we also used the unit circle at the end to determine the values of cos π/6 and 5cos π/6. Therefore, the area of the shaded region is: But what should the limits of integration be? We first have to solve sin x = 1/2.īased on the unit circle, there are two angles at which the sine value is 1/2: π/6 and 5π/6. We know that a properly set-up integral will do the trick. Using the Unit Circleįind the area of the shaded region between y = sin x and the line y = 1/2. Now let’s see how we can use these identities in actual AP Calculus problems. In fact, there are just a few key items to remember. So if sine-squared or cosine-squared shows up in an integral, remember these identities.Īs you can see, the list of essential trig identities is not terribly long. The half-angle identities can be used to convert a squared expression into a form that is easier to deal with. When applied to the unit circle, we get a trio of useful trigonometric identities: If the side lengths of a right triangle are a and b, and the hypotenuse is c, then: Perhaps the most famous and useful equation in all of mathematics is the Pythagorean Theorem. The following rules provide shortcuts for dealing with negative input.

The quotient identities are useful for re-expressing the trig functions in terms of sin and/or cos. Below is a list of what I would consider the essential identities. The Essential Trigonometric Identitiesįortunately, you do not have to remember absolutely every identity from Trig class.

The Unit Circle, which displays cosine and sine values of common angles.

Calc trig identities cheat sheet