Find the exact value of $\sin^{-1}\left (-\frac{\sqrt{3}}{2}\right)$ and the like.
a. $\sin^{-1} \left(-\frac{\sqrt{3}}{2}\right)$
b. $\cot^{-1} \left(-1\right)$
c. $\sec^{-1} \left(\frac{2}{\sqrt{3}}\right)$
d. $\cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$
e. $\tan^{-1} \left(-\sqrt{3}\right)$
How do you do these questions without a calculator? You don't have to do them all—I just need to know how to do at least one of them! I appreciate any advice and/or solutions.
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$\begingroup$There are a few values of $\sin$ or $\cos$ that you should know, e.g. $\sin(\pi/3) = \sqrt{3}/2$. Using the fact that $\sin$ is an odd function, $\sin(-\pi/3) = -\sqrt{3}/2$. So $\sin^{-1}(-\sqrt{3}/2) = -\pi/3$.
$\endgroup$ 4 $\begingroup$An equilateral triangle has all angles equal to $60^\circ$. Let this triangle have a side length of $2$. Divide the triangle into two right triangles. One of the legs of the right triangle is of length $1$, the hypotenuse is of length $2$, and from the Pythagorean theorem, the other leg is of length $\sqrt{3}$. From this we get $\sin(60^\circ)=\frac{\sqrt{3}}{2}$. You can use this triangle to find the values of the trigonometric functions of $30^\circ$ and $60^\circ$.
For trigonometric functions of $45^\circ$, use an isosceles right triangle.
$\endgroup$ $\begingroup$There are easy-to-remember vales of the sine, cosine and tangent functions for 0, 30, 45, 60 and 90 degrees:
- Sine: 0, $\frac12$, $\frac{\sqrt2}2$, $\frac{\sqrt3}{2}$, 1 respectively
- Cosine: reverse the above sequence of values
- Tangent: 0, $\frac1{\sqrt3}$, 1, $\sqrt3$, $\infty$
Then you need to remember how each of their graphs looks like, their associated symmetry relations and their reciprocal functions. Then you can work out the values of these inverse functions you have here.
For example, the first question asks for x where $\sin x=-\frac{\sqrt3}2$. Since the sine function is odd, $\sin(-x)=\frac{\sqrt3}2$ and so x is $-60°$ ($-\frac\pi3$ radians).
$\endgroup$ 3 $\begingroup$You are clear on how to do these questions. Therefore in this answer, no explanation is necessary. Solely answers will be provided. Here there are:
a. $-\frac{\pi}{3}$
b. $-\frac{\pi}{4}$
c. $\frac{\pi}{6}$
d. $\frac{3\pi}{4}$
e. $\frac{5\pi}{3}$
In addition, is a very useful website only if you want simple calculations such as the ones above. Feel free to use it! (You need to upgrade membership in order to view solutions, however.)
NOTE: If one of the answers above happen to be incorrect, can someone be kind enough to notify me? Thank you very much!
Good luck, @CGuan! :)
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