Take x = cos θ , then proceeding as above, we get, sin – ( = cos – x Example Express in the simplest form. Solution We write 2sin = – cos – tan tan tan INVERSE TRIGONOMETRIC FUNCTIONS Example Write , x > in the simplest form. Solution Let x = sec θ , then x − = sec θ −= θ Therefore, x − = cot – (cot θ ) = θ = sec – x, which is the simplest form. EXERCISE .
Prove the following: . 3sin – x = sin – ( x – x ), – ∈ . 3cos – x = cos – ( x – x ), , ∈ Write the following functions in the simplest form: . − , x ≠ .
− , < x < π . cos − , −π < x < . a , | x | < a . a x a ax − , a > ; a a Find the values of each of the following: .
– 2cos 2sin . y y , | x | < , y > and xy < Find the values of each of the expressions in Exercises to . . sin .
tan . tan sin . is equal to (A) (B) (C) (D) . ( is equal to (A) (B) (C) (D) .
( ) is equal to (A) π (B) (C) (D) Miscellaneous Examples Example Find the value of (sin Solution We know that (sin ) . Therefore, (sin But ∉− , which is the principal branch of sin – x However sin ( sin( π − and ∈− Therefore (sin (sin INVERSE TRIGONOMETRIC FUNCTIONS Miscellaneous Exercise on Chapter Find the value of the following: . cos . tan Prove that .
Prove that . , x ∈ [ , ] . , ∈ . ≤ [Hint: Put x = cos θ ] Solve the following equations: .
2tan – (cos x ) = tan – ( cosec x ) . ,( ) x x > . sin (tan – x ), | x | < is equal to (A) (B) (C) (D) . sin – ( – x ) – sin – x = π , then x is equal to (A) , (B) , (C) (D) Summary