Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Find the exact value of $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )$ without using a calculator.

I started by finding $\sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )=\dfrac{\pi}{4}$

So, $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )=\tan\left( \dfrac{\pi}{4}\right)$.

The answer is $1$. Can you show how to solve $\tan\left( \dfrac{\pi}{4}\right)$ to get $1$? Thank you.

share|cite|improve this question
So you did all that but you can't find $\tan\left( \dfrac{\pi}{4}\right)$? – Git Gud Jan 1 '14 at 2:01
Draw an isoceles right-angled triangle. Two of its angles are $\pi/4$. Then $\tan(\pi/4)$ is opposite divided by adjacent. These are equal, so $\tan(\pi/4)=1$. – André Nicolas Jan 1 '14 at 2:07
You don't have to actually compute $\sin^{-1}(\sqrt{2}/2)$ to solve this problem. Draw a right triangle with opposite side length $\sqrt{2}$ and hypotenuse length $2$. Now use the pythagorean theorem to find the length of the adjacent side; call this length $a$. Then compute $\sqrt{2}/a$. – Alex Wertheim Jan 1 '14 at 2:12

4 Answers 4

up vote 5 down vote accepted

Hint: $$\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)}$$

share|cite|improve this answer
You mispelled 'answer'. – Git Gud Jan 1 '14 at 2:05
No, because he didn't also misspell $1$. – John Jan 1 '14 at 2:14

$\hskip2in$ enter image description here

Using the triangle above...& the fact that $$\tan x = \frac{\text{opp}}{\text{adj}}, \space \tan \left(\frac{\pi}{4}\right)=...$$

share|cite|improve this answer

$tan(sin^{-1}\frac{1}{\sqrt{2}})$. So $sin(45)=\frac{1}{\sqrt{2}},x=45$ so $tan(45)=1$ hence done . Now $sin(45)=cos(45)$ thus $tan(45)=1$

share|cite|improve this answer

You have

$ \sin x=\frac{\sqrt{2}}{2} $


$ \cos x=\sqrt{1-(\sin x)^{2}}=\frac{\sqrt{2}}{2} $


$ \tan x =\frac{\sin x}{\cos x}\ = 1 $

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.