How to Solve Inverse Trig Functions Without a Calculator?

Click For Summary
SUMMARY

This discussion focuses on solving inverse trigonometric functions without a calculator, specifically examples like sec(arctan 2) and cos(2arcsin(5/13). The method involves constructing right triangles based on the definitions of the trigonometric functions. For sec(arctan 2), a right triangle is drawn with the opposite side labeled as 2 and the adjacent side as 1, allowing the hypotenuse to be calculated using the Pythagorean theorem. This approach enables the calculation of sec(A) directly from the triangle's sides.

PREREQUISITES
  • Understanding of inverse trigonometric functions
  • Knowledge of right triangle properties
  • Familiarity with Pythagorean theorem
  • Basic trigonometric identities
NEXT STEPS
  • Study the derivation of trigonometric identities
  • Practice constructing right triangles for various inverse trigonometric functions
  • Learn how to apply the Pythagorean theorem in trigonometric contexts
  • Explore more complex inverse trigonometric problems without calculators
USEFUL FOR

Students, educators, and anyone interested in mastering trigonometry, particularly those looking to solve inverse trig functions manually without relying on calculators.

gech
Messages
4
Reaction score
0
Supposing I need to solve a problem like: sec(arctan 2) or cos(2arcsin(5/13)), is there a method I could use that would not require a calculator? What I mean is that for an example like tan(arccos .5), the answer is "simple" because I know the arc cosine of .5 is pi/3 and then the tan of pi/3 is the squareroot of 3. But in a problem like the two above, how would I go about doing this where the numbers are more complicated and I want to do it by hand? Or is it impossible without the use of a calculator?
 
Physics news on Phys.org
Here's one way: In the case of sec(atan(2)) draw a right triangle and label one of the angles, say A. Label the side opposite A with 2 and the side adjacent to it with 1. Clearly, A is an angle whose tangent is 2. What's the secant of this angle? Use Pythagoras to find the length of the hypotenuse and divide it by 1 (adjacent side) to find sec(A) = sec(atan(2)).

You can do similar things for your other example by applying well known trig identities.
 
I see what you're getting at. Thank you.
 

Similar threads

Undergrad Why is the integral of ##\arcsin(\sin(x))## so divisive?
  • · Replies 3 ·
Replies
3
Views
3K
How to Determine Correct Inverse Trig Angle?
  • · Replies 2 ·
Replies
2
Views
1K
High School Advice to obtain the domain of compound functions
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Inverse trigonometric functions
  • · Replies 1 ·
Replies
1
Views
1K
MHB How can I find the inverse function for a given approximation?
  • · Replies 3 ·
Replies
3
Views
3K
MHB How do you solve this calculus 2 integration problem?
  • · Replies 13 ·
Replies
13
Views
4K
Converting one inverse trig function to another
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Calculating the inverse of a function involving the error function
  • · Replies 3 ·
Replies
3
Views
2K
Uniform Continuity of functions
  • · Replies 40 ·
2
Replies
40
Views
6K
Graduate How to solve an integral with the Inverse error function
  • · Replies 5 ·
Replies
5
Views
4K