Trigonometric Values Exact Values

In summary, the problem involves finding exact values for inverse trigonometric functions. The first one is solved by using the cotangent function and Pythagoras' theorem to find the adjacent side. The same method can be used for the other two questions, but with the use of known identities and symmetries for negative values. The correct answers for 1b and 1c are √34/5 and 4/√15, respectively.
  • #1
ardentmed
158
0
Hey guys,

I've a few more questions this time around from my problem set:

(Ignore the first question , I only need help with 2a, b,and c.)

Question:
08b1167bae0c33982682_5.jpg


For the first one, I assessed the inside of the inverse trigonometric function first:

sin^-1 (2/3) = 0.785
Then tan-1(.785) ~ .66577,

Likewise, for b, I got 2.14173 and undefined for c.

However, I am having trouble finding these in exact values. How would I go about doing that? For example, is sin^-1 (2/3) a common trigonometric value I should be committing to memory? Thanks again guys.
 
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  • #2
Let's look at the first one, and then you can try the others...

\(\displaystyle \sin^{-1}\left(\frac{2}{3}\right)\)

can represent an angle in a right triangle whose opposite side is 2 and the hypotenuse is 3. So, we need to find the adjacent side $x$, and by Pythagoras we find:

\(\displaystyle x=\sqrt{3^2-2^2}=\sqrt{5}\)

Since the cotangent function is defined as adjacent over opposite, we mat then state:

\(\displaystyle \cot\left(\sin^{-1}\left(\frac{2}{3}\right)\right)=\frac{\sqrt{5}}{2}\)

Can you now use a similar method for the others? With the negative values, use known identities and symmetries.
 
  • #3
MarkFL said:
Let's look at the first one, and then you can try the others...

\(\displaystyle \sin^{-1}\left(\frac{2}{3}\right)\)

can represent an angle in a right triangle whose opposite side is 2 and the hypotenuse is 3. So, we need to find the adjacent side $x$, and by Pythagoras we find:

\(\displaystyle x=\sqrt{3^2-2^2}=\sqrt{5}\)

Since the cotangent function is defined as adjacent over opposite, we mat then state:

\(\displaystyle \cot\left(\sin^{-1}\left(\frac{2}{3}\right)\right)=\frac{\sqrt{5}}{2}\)

Can you now use a similar method for the others? With the negative values, use known identities and symmetries.
Thanks, I got the same answer for the first one after drawing the triangles out. For the second and third one, I got 5/√ ̅34 and √ ̅15 / 4. Am I on the right track?
 
  • #4
The second one is incorrect (2b)...but without seeing your work, I can't tell you where you went wrong.
 
  • #5
I suspect he's misreading the triangle. \(\displaystyle sec\theta = \frac{H}{A}\), h is the hypotenuse, and a is the adjacent. 1c) is also wrong for the same reason.
 
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  • #6
Rido12 said:
I suspect he's misreading the triangle. \(\displaystyle sec\theta = \frac{H}{A}\), h is the hypotenuse, and a is the adjacent.

Yes, I suspect that as well, but I am trying to discourage the habit of simply posting an answer and saying "is that right?" I want to see work and reasoning, this way I can point to where the error is if the result is wrong. :D
 
  • #7
Rido12 said:
I suspect he's misreading the triangle. \(\displaystyle sec\theta = \frac{H}{A}\), h is the hypotenuse, and a is the adjacent. 1c) is also wrong for the same reason.

Oh, my bad. Alright, well since drawing out the triangles on a computer is such a hassle, I'll just post the new answers:

I got √ ̅34 /5and 4/√ ̅15

I'm hoping these are right. Thanks guys!
 
  • #8
Those are correct, but if you really want to, you can rationalize the denominator for 1c).
 

FAQ: Trigonometric Values Exact Values

What are Trigonometric Values Exact Values?

Trigonometric values exact values are the values of the sine, cosine, and tangent functions for specific angles that can be expressed without using a calculator or approximations.

Why are Trigonometric Values Exact Values important?

Trigonometric values exact values are important because they allow for accurate calculations in trigonometry and other fields such as engineering and physics.

How do you find Trigonometric Values Exact Values?

Trigonometric values exact values can be found using special triangles, unit circle, or trigonometric identities and formulas.

What are the most commonly used Trigonometric Values Exact Values?

The most commonly used Trigonometric values exact values are 0, 1/2, 1/√2, √2/2, √3/2, and 1, for angles such as 0°, 30°, 45°, 60°, and 90° respectively.

How do Trigonometric Values Exact Values differ from Trigonometric Approximate Values?

Trigonometric values exact values are the exact values of trigonometric functions for specific angles, while trigonometric approximate values are rounded or estimated values using a calculator or other methods.

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