Trigonometric identities are equations that involve trigonometric functions such as sine, cosine, and tangent, and are true for all values of the variables involved. They are used to simplify and manipulate trigonometric expressions.
Understanding trigonometric identities allows us to simplify complex trigonometric expressions and solve trigonometric equations more easily. It also helps in proving mathematical theorems and in applications of trigonometry, such as in physics and engineering.
The sum and difference identities are trigonometric identities that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the individual angles. They are: sin(A ± B) = sinAcosB ± cosAsinB, cos(A ± B) = cosAcosB ∓ sinAsinB, and tan(A ± B) = (tanA ± tanB) / (1 ∓ tanAtanB).
To use sum and difference identities, you need to know the values of the trigonometric functions for the individual angles. Then, you can plug in those values into the respective identity to find the value of the trigonometric function for the sum or difference of the angles.
Multiple angle identities are trigonometric identities that express the sine, cosine, and tangent of an angle multiplied by a number (such as 2, 3, or 4) in terms of the sine, cosine, and tangent of the original angle. They are: sin(nA) = 2sinAcosA cos(nA) = cosnA - sinnA, and tan(nA) = (tanAn - n tanA) / (1 - tan2An).