Solving Eq & Ineq        Graphs & Func.        Systems of Eq.        Polynomials        Frac. Express.        Powers & Roots        Complex Numbers        Quadratic Eq.        Quadratic Func.       Coord. Geo.        Exp. & Log. Func.        Probability        Matrices        Trigonometry    Trig. Identities        Equations & Tri. On this page we hope to clear up problems that you might have with the trigonometric identities, such as the double angle identities and the half-angle identities.  Scroll down or click any of the links below to better your understanding of the trig. identities. Cosines Other identities Double angle identities Proving identities Quiz on Trigonometric Identities The trig. identities are important identities that involve sums or differences of angles.  An identity that shows that the cosine of the difference of two angles is related to the cosines and sines of the angles themselves.  This identity is given below (A and B are used in place of alpha and beta, respectively since HTML does not support Greek characters). cos (A - B) = (cos A)(cos B) + (sin A)(sin B) The figure below gives a graphical representation of the cosine identity. This identity is useful when you are asked to find the cosine of a non-30o-45o-60o-90o angle, such as 10o.  Example: ``` 1. Problem: Find cos 15o. Solution: Write 15o in terms of angles with known trig. ratio values. cos (45o - 30o) Use the cosine identity to rewrite the expression. (cos 45o)cos 30o + (sin 45o)sin 30o Using the values you know for the trig. ratios of special angles, rewrite the expression. SQRT(2) SQRT(3) SQRT(2) 1 ------- * ------- + ------- * - 2 2 2 2 Perform the indicated multiplications. SQRT(6) SQRT(2) ------- + ------- 4 4 SQRT(6) + SQRT(2) ----------------- 4 ``` There is also a cosine identity for a sum of angles.  It is shown below. cos (A + B) = (cos A)cos B - (sin A)sin B There are also sine identities and tangent identities.  They are listed below. sin (A + B) = (sin A)(cos B) + (cos A)(sin B) sin (A - B) = (sin A)(cos B) - (cos A)(sin B) tan (A + B) = (tan A + tan B)/(1 - (tan A)(tan B)) tan (A - B) = (tan A - tan B)/(1 + (tan A)(tan B)) Example: ``` 1. Problem: Find tan 15o. Solution: Rewrite as a difference of angles with known trig. ratio values. tan (45o - 30o) Use the tangent of differences identity to rewrite the expression. tan 45o - tan 30o ---------------------- 1 + (tan 45o)(tan 30o) Substitute the known trig. ratio values in and perform the indicated operations. 1 - (SQRT(3))/3 --------------- 1 + (SQRT(3))/3 3 - SQRT(3) ----------- 3 + SQRT(3) 2 - SQRT(3)``` Identities involving sin 2x of cos 2x are called double-angle identities.  These identities are derived using the sum and difference identities.  Below, we will show you how one of the double-angle identities was derived. ``` sin 2x = sin (x + x) Use the sine of sums identity. (sin x)(cos x) + (cos x)(sin x) 2(sin x)(cos x)``` Example: ``` 1. Problem: If the sine of theta is (3/8) and theta is in the first quadrant, what is sin 2(theta). (Use the figure below.) ``` ``` Solution: From the diagram, we see that cos (theta) = (SQRT(55)/8. Use the double angle identity for sine. sin 2(theta) = 2(sin (theta))(cos (theta)) Plug in the values you know. 2 * (3/8) * (SQRT(55)/8) Perform the indicated multiplications. (3(SQRT(55)))/32``` The other identities are listed below. cos 2x = cos2 x - sin2 x cos 2x = 1 - 2sin2 x cos 2x = 2cos2 x - 1 tan 2x = (2tan x)/(1 - tan2 x) sin2 x = (1 - cos 2x)/2 cos2 x = (1 + cos 2x)/2 Identities can be used to prove identities equal to other identities.  Example: ``` 1. Problem: Prove the following identity: tan2 x - sin2 x = (sin2 x)(tan2 x) Solution: Write each side in terms of sin x and cos x. sin2 x sin2 x ------ - sin2 x = (sin2 x)------ cos2 x cos2 x Now, only deal with one side of the equation. Find the common denominator and subtract. sin2 x - (sin2 x)(cos2) ----------------------- cos2 x Factor out a sin2 x. (sin2 x)(1 - cos2 x) -------------------- cos2 x Use the Pythagorean Identities to replace (1 - cos2 x) with sin2 x. (sin2 x)(sin2 x) ---------------- cos2 x sin2 x (sin2 x)------ cos2 x``` Take the Quiz on trigonometric identities.  (Very useful to review or to see if you've really got this topic down.)  Do it!

Math for Morons Like Us - Algebra II: Trigonometric Identities
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