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An inverse function that works in reverse exists for every trigonometric function. All these trigonometric functions start with “arc” in their names. Even the secant has a reverse name and it goes by arcsec.
We know that the first quadrant is between 0 degrees and 90 degrees. In this quadrant, the values of sine, cosine, and tangent are positive. Besides that, we also know that the right angle is always equal to 90 degrees.
sin-1 ( – y ) = – sin -1 ( y ), y Є [ – 1, 1 ]
cos-1 ( – y ) = pi* – cos -1 ( y ), y Є [ – 1, 1 ]
tan-1 ( – y ) = – tan -1 ( y ), y Є R
cosec-1 ( – y ) = – cosec -1 ( y ), | y | ≥ 1
sec -1 ( – y ) = pi – sec -1 ( y ), | y | ≥ 1
cot -1 ( – y ) = pi – cot -1 ( y ), y Є R
sin-1 ( – y ) + cos-1 ( – y ) = pi /2 , y Є [ – 1, 1 ]
tan-1 ( – y ) + cot -1 ( – y ) = pi /2 , y Є R
sec -1 ( – y ) + cosec-1 ( – y ) = pi /2 , | y | ≥ 1
Sin -1 ( 1 / y ) = cosec -1 ( y ), if y ≥ 1 or y ≤ – 1
cos -1 ( 1 / y ) = sec -1 ( y ), if y ≥ 1 or y ≤ – 1
tan -1 ( 1 / y ) = cot -1 ( y ), if y > 0
tan-1 x + tan-1 y = tan-1 ((x + y) / (1 – xy)), if the value xy < 1
tan-1 x – tan-1 y = tan-1 ((x – y) / (1 + xy)), if the value xy > – 1
2 tan -1 y = sin -1 ( 2x / ( 1 + x2 )), | x | ≤ 1
2 tan -1 y = cos-1 (( 1- y2)/(1+ y2)), y ≥ 0
2 tan-1 = tan-1 ( 2y / (1-y2 )), -1 < y < 1
3sin-1 y = sin-1 ( 3x – 4y3)
3cos-1 y = cos-1 ( 4y3 -3y )
3tan-1 y = tan-1 (( 3y – y3 ) / (1 – 3y2))
sin(sin-1 ( y )) = y, -1 ≤ y ≤ 1
cos ( cos-1 ( y ) ) = y, -1≤ y ≤1
tan (tan-1 ( y )) = y , – ∞ < x < ∞
cosec ( cosec-1 ( y )) = y, – ∞ < y ≤ 1 or -1 ≤ y < ∞
sec (sec-1 ( y )) = y,- ∞ < y ≤ 1 or 1 ≤ y < ∞
Question 1: Find the value of tan y if secant y is equal to 1.