All Precalculus Resources
Example Questions
Example Question #1 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Approximate:
:
There is a restriction for the range of the inverse tangent function from .
The inverse tangent of a value asks for the angle where the coordinate lies on the unit circle under the condition that
. For this to be valid on the unit circle, the
must be very close to 1, with an
value also very close to zero, but cannot equal to zero since
would be undefined.
The point is located on the unit circle when
, but
is invalid due to the existent asymptote at this angle.
An example of a point very close to that will yield
can be written as:
Therefore, the approximated rounded value of is
.
Example Question #31 : Graphs And Inverses Of Trigonometric Functions
Evaluate:
To determine the value of , solve each of the terms first.
The inverse cosine has a domain and range restriction.
The domain exists from , and the range from
. The inverse cosine asks for the angle when the x-value of the existing coordinate is
. The only possibility is
since the coordinate can only exist in the first quadrant.
The inverse sine also has a domain and range restriction.
The domain exists from , and the range from
. The inverse sine asks for the angle when the y-value of the existing coordinate is
. The only possibility is
since the coordinate can only exist in the first quadrant.
Therefore:
Example Question #3 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Evaluate the following:
For this particular problem we need to recall that the inverse cosine cancels out the cosine therefore,
.
So the expression just becomes
From here, recall the unit circle for specific angles such as .
Thus,
.
Example Question #1 : Inverse Trigonometric Functions
Determine the value of in degrees.
Rewrite and evaluate .
The inverse sine of one-half is since
is the y-value of the coordinate when the angle is
.
To convert from radians to degrees, replace with 180.
Example Question #5 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Evaluate the following expression:
This one seems complicated, but becomes considerably easier once you implement the fact that the composite cancels out to 1 and you are left with
which is equal to 1
Example Question #851 : Pre Calculus
Evaluate:
Example Question #852 : Pre Calculus
Approximate the following:
This one is rather simple with knowledge of the unit circle: the value is extremely close to zero, of which always
Example Question #8 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Given that and that
is acute, find the value of
without using a calculator.
Given the value of the opposite and hypotenuse sides from the sine expression (3 and 4 respectively) we can use the Pythagorean Theorem to find the 3rd side (we’ll call it “t”): . From here we can easily deduce the value of
(the adjacent side over the opposite side)
Example Question #9 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Given that and that
is acute, find the value of
without using a calculator.
Given the value of the opposite and hypotenuse sides from the sine expression (3 and 4 respectively) we can use the Pythagorean Theorem to find the 3rd side (we’ll call it “t”): .
From here we can deduce the value of (the adjacent side over the opposite side) and so the answer is
.
Example Question #2 : Evaluate Expressions That Include The Inverse Tangent, Cosecant, Secant, Or Cotangent Function
Evaluate the following expression:
This one seems complicated but becomes considerably easier once you implement the fact that the composite cancels out to
and you are left with
which is equal to
, and so the answer is
.
All Precalculus Resources
![Learning Tools by Varsity Tutors](https://vt-vtwa-app-assets.varsitytutors.com/assets/problems/og_image_practice_problems-9cd7cd1b01009043c4576617bc620d0d5f9d58294f59b6d6556fd8365f7440cf.jpg)