How to Solve Inverse Trig Functions Without a Calculator?

In summary, the conversation discusses methods for solving trigonometric problems without the use of a calculator. It suggests using right triangles and known trig identities to find the solution. The method is demonstrated with two examples - sec(arctan 2) and cos(2arcsin(5/13)). It is possible to solve these problems by hand with the help of trigonometric concepts.
  • #1
gech
4
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?
 
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  • #2
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.
 
  • #3
I see what you're getting at. Thank you.
 

FAQ: How to Solve Inverse Trig Functions Without a Calculator?

1. What are inverse trig functions?

Inverse trig functions are mathematical operations that undo the effects of trigonometric functions. They help find the angle or side length in a right triangle when given the ratio of two sides.

2. Why are inverse trig functions important?

Inverse trig functions are important because they allow us to solve for unknown angles or sides in right triangles, which is useful in many real-life applications such as navigation, engineering, and physics.

3. How do you solve inverse trig functions?

To solve inverse trig functions, you need to use the inverse trigonometric ratios (arcsin, arccos, arctan) along with the given ratio to find the missing angle or side length in a right triangle. You can use a calculator or refer to a trigonometric table to find the value of the inverse trig function.

4. What is the range of inverse trig functions?

The range of inverse trig functions depends on the particular function. The range of arcsine and arccosine functions is between -π/2 and π/2 (in radians) or -90° and 90° (in degrees). The range of arctangent function is between -π/2 and π/2 (in radians) or -90° and 90° (in degrees).

5. What are some common mistakes when solving inverse trig functions?

Some common mistakes when solving inverse trig functions include forgetting to convert between degrees and radians, using the wrong inverse trig function, and not checking for possible multiple solutions. It is important to carefully read the problem and pay attention to the given information to avoid these mistakes.

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