Find the exact value of each expression
Cos^-1(0)
Tan^-1 sqrt(3)/3
Sin^-1(1)
Since the expressions are capitalized, I'm not sure if I have to do anything different. Any help is appreciated.
I just noticed that this remained unanswered.
the caps really have no special significance.
for the first and last, look at the sine and cosine curves.
Where does the cosine curve cross the x-axis ? ---- at pi/2
so cos^-1 (0) = pi/2
similarly where is the sine curve equal to 1 ?
at pi/2
for Tan^-1 sqrt(3)/3
look at a right-angled triangle with height of √3 and base of 3
- with your calculator set to radians, divide √3 by 3 then press
2ndF TAN (or INV TAN) to get .5236 radians which is pi/6 radians or ...
repeating the sequence again in degrees
30º
After a closer look we should realize that √3/3 = 1/√3
so our triangle would have a height of 1, a base of √3, and a hypotenuse of 2
which of course are the ratios of the
30º, 60º, 90º triangle.
2tan-sin
To find the exact value of each expression, we can use the inverse trigonometric functions.
1. Find the exact value of `cos^-1(0)`:
The expression `cos^-1(0)` represents the inverse cosine function applied to 0. This means we need to find the angle whose cosine is 0.
The cosine of an angle is equal to 0 at two points in the unit circle: at π/2 radians (90 degrees) and at 3π/2 radians (270 degrees). Therefore, the exact value of `cos^-1(0)` is π/2 or 3π/2.
2. Find the exact value of `tan^-1(sqrt(3)/3)`:
The expression `tan^-1(sqrt(3)/3)` represents the inverse tangent function applied to the ratio of the square root of 3 and 3.
To find the exact value, we can rationalize the denominator by multiplying both numerator and denominator by √3:
tan^-1(sqrt(3)/3) = tan^-1((sqrt(3)/3) * (√3/√3))
= tan^-1(sqrt(3) * √3 / 3√3)
= tan^-1(sqrt(3 * 3) / 3√3)
= tan^-1(√9 / 3√3)
= tan^-1(3 / 3√3)
= tan^-1(1 / √3)
Using the special right triangle with angles 30 degrees, 60 degrees, and 90 degrees, we know that tan(30 degrees) = 1/√3. Therefore, the exact value of `tan^-1(sqrt(3)/3)` is 30 degrees.
3. Find the exact value of `sin^-1(1)`:
The expression `sin^-1(1)` represents the inverse sine function applied to 1. This means we need to find the angle whose sine is 1.
The sine of an angle is equal to 1 at π/2 radians (90 degrees). Therefore, the exact value of `sin^-1(1)` is π/2 or 90 degrees.