Find the exact value of the inverse trig functions.?

Question

Find the exact value of the inverse trig functions.

(1) csc[cos^1 (-(sqrt{3}/2)]

(2) tan[sin^1(-(1/2)]

Answer 1

1) csc[cos^-1 (-(sqrt{3}/2)]

let x=cos^-1 (-(sqrt{3}/2)

cos x=-(sqrt{3}/2

x=120 or 330
1/sin 120= 1/sqrt(3)/2=2 sqrt(3)/3


do the other in the same way

Answer 2

csc[cos^1 (-(sqrt{3}/2)]

This is asking you for the cosecant of the angle whose cosine is negative square root of 3 divided by 2 and lies in the interval 0 to pi (recall that this the range of the inverse cosine). Thus the angle whose cosine satisfies both of these conditions is 5pi/6 (or 150 degrees). Moving on,
csc (5pi/6) is the same as 1/sin(5pi/6), and since
sin(5pi/6)=1/2, the value of the given expression is 2.


tan[sin^1(-(1/2)]

This is asking you for the tangent of the angle whose sine is negative 1/2 and lies in the interval –pi/2 to pi/2 (recall that this the range of the inverse sine). Thus the angle whose sine satisfies both of these conditions is –pi/6
(or - 30 degrees). Moving on, tan (-pi/6) is negative square root of 3 divided by 3, thereby giving you the value of the given expression.

The solution presented by Mathematica above provides another way of evaluating the expressions based on the right triangle definitions of the trig functions.

Answer 3

often, the inverse function of a function supplies the cost of the argument of the function. yet Sec^-a million would not provide you a different value. discover ?=7?/6 on your unit circle graph, and observe that the cosine of 5?/6 is comparable to the cosine of 7?/6. the alternative/hypotenuse value are an identical using fact the secant is basically the inverse of the cosine, 5?/6 and 7?/6 are the two non-trivial ideas for the equation y=Sec^-a million(sec((7pi)/6).

Answer 4

2)
Since the cos is negative, the triangle is in the second quadrant.

cos = adj / hyp

The adjacent leg (horizontal) is -sqrt 3
The hypotenuse is 2
Using pythagorean theorem
(hyp)^2 = (opp)^2 + (adj)^2
you find that the opposite leg (vertical) is 1

So, now find the CSC of that angle.
csc x = hyp / opp = 2 / 1 = 2

2)
Since sin is negative, the triangle is in the fourth quadrant.

sin = opp / hyp

So, we know that
opposite = -1
hypotenuse = 2

using pythagorean theorem (like above), we find that
adjacent = sqrt 3

tan = opp / adj = -1 / sqrt 3 = (-sqrt 3) / 3