Finding the sine, cos of common angles without calculator

• Nov 1st 2012, 05:53 AM
ScorpFire
Finding the sine, cos of common angles without calculator
Hi,

I have heard its possible to find the sine and cosine of some common angles without calculator.

Can anyone explain how please?

eg.

cos pi/4 = 1/√2
sin pi/4 = 1/√2

cos 5pi/4 = -1/√2

cos pi/3 = 1/2

sin pi/3 = √3/2

• Nov 1st 2012, 06:29 AM
BobP
Re: Finding the sine, cos of common angles without calculator
Look at the thread when tanx = 1
• Nov 1st 2012, 07:06 AM
Soroban
Re: Finding the sine, cos of common angles without calculator
Hello, ScorpFire!

Quote:

I have heard its possible to find the sine and cosine of some common angles without calculator.

Can anyone explain how please?

For example: . $\begin{array}{cccccccccccc}\sin\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} && \sin\frac{\pi}{3} \:=\:\frac{\sqrt{3}}{2} && \sin\frac{\pi}{6} \:=\:\frac{1}{2} \\ \\[-3mm] \cos\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} && \cos\frac{\pi}{3} \:=\:\frac{1}{2} && \cos\frac{\pi}{6} \:=\:\frac{\sqrt{3}}{2} \end{array}$

$\text{For }\theta = \tfrac{\pi}{4}\:(45^o)$, consider an isosceles right triangle.
Code:

      *       * *       *  *  h     1 *    *       *      *       *      45 *       *  *  *  *  *             1
Let the equal sides equal 1.
Pythagorus says the hypotenuse is $\sqrt{2}.$

We have: . $\begin{Bmatrix} opp &=& 1 \\ adj &=& 1 \\ hyp &=& \sqrt{2}\end{Bmatrix}$
And you can write the trig values of $\tfrac{\pi}{4}$

Memorize "one, one, square-root-of-two".

$\text{For }\theta =\tfrac{\pi}{3}\:(60^o)$, consider an equilateral triangle
. . with side length 2. .Draw an altitude.
Code:

            *           *|*           * | *       2 *  |y *         *  |  *       * 60 |    *       *  *  *  *  *       :  1  :  1  :
We have: . $adj = 1,\:hyp = 2$
Pythagorus says: $opp = \sqrt{3}$
And you can write the trig values for $\tfrac{\pi}{3}$

For $\theta = \tfrac{\pi}{6}\;(30^o)$, turn the above right triangle on its side.
Code:

                  *           2    *  *             *    * 1         * 30    *       *  *  *_ *  *             √3
We have: . $\begin{Bmatrix}opp &=& 1 \\ adj &=& \sqrt{3} \\ hyp &=& 2\end{Bmatrix}$
And you can write the trig values for $\tfrac{\pi}{6}$

For both diagrams, memorize "one, two, square-root-of-three".

Be careful! .Remember that:
. . The shortest side, 1, is opposite the smallest angle, 30o.
. . The longest side, 2, is opposite the largest angle, 90o.
• Nov 1st 2012, 07:50 AM
richard1234
Re: Finding the sine, cos of common angles without calculator
Take a point on the unit circle. The cosine of the corresponding angle is simply the x-coordinate (adjacent over hypotenuse). The sine of that angle is the y-coordinate. That's how I usually find sine/cosine in my head by just visualizing the unit circle.
• Nov 2nd 2012, 10:14 PM
Salahuddin559
Re: Finding the sine, cos of common angles without calculator
There are formulae for finding sin(A+B) and cos(A+B) etc... using these one can extend the values for other angles. For example, sin2A = 2sinAcosA, using which one can find sin(15 degrees) = sqrt(2 - sqrt(3))/2.

Salahuddin
Maths online