How do you evaluate sine, cosine, tangent of pi/4 without using a calculator?

2 Answers
Dec 14, 2016

$\pi$ means 180 °

Explanation:

Hence $\frac{\pi}{4}$ means $\frac{180}{4}$ which is 45 ° .

45 °  is a special triangle where the measurements are

Now, you can find the sine, cosine and tangent without a calculator!

You should obtain

Sin and cosine: $\frac{1}{\sqrt{2}}$
Tangent: $1$

I am assuming you are familiar with the formula SOH CAH TOA, yes?

Dec 14, 2016

Start by drawing a right triangle in Quadrant I, with a ${45}^{\circ}$ angle, which is the equivalent angle for $\frac{\pi}{4}$.

Explanation:

The ${45}^{\circ} - {45}^{\circ} - {90}^{\circ}$ right triangle has legs of equal length, so the ratio for tangent = 1. (opposite over adjacent)

I teach that ${x}^{2} + {x}^{2} = 1$ is the Pythagorean Theorem for the isosceles right triangle with a hypotenuse = 1. To solve, you get $2 {x}^{2} = 1$, so ${x}^{2} = \frac{1}{2}$.
Take the square root of both sides:
$x = \pm \frac{\sqrt{2}}{2}$

If each side = $\frac{\sqrt{2}}{2}$, then both the sine and cosine are equal to that value since the hypotenuse is 1.