Examples NOT using cosine rule

If the following questions came up in the exam we recommend using LOGIC to work out how to solve this problem using standard trigonometry (SOH CAH TOA) and Pythagoras.

Example 1

What is the length Q to R in the following diagram?

What is the length of Q to R

To solve this redraw the diagram as:

Redraw the diagram

Use trigonometry (SOH CAH TOA) to find the value of $x$

Find value of x using trigonometry

$costheta=a/H$

$cos35^@=x/3$

$x=cos35^@times3$

$x=0.8191times3=2.457$

$x=2.457$

Now use Pythagoras’s theorem to find $Y$

Find value of Y using Pythagoras

$H^2=x^2+Y^2$

$3^2=2.457^2+Y^2$

$Y^2=3^2-2.457^2$

$Y^2=9-6.0368$

$Y^2=2.9631$

$Y=sqrt 2.9631$

$Y=1.721$

Now find $z$

Find the value of X

$z$ is simply $7-2.457=4.543$

$z=4.543$

Then use Pythagoras’s theorem to find QR

Find the value of QR using Pythagoras theorem

$H^2=x^2+y^2$

$QR^2=4.543^2+1.721^2$

$QR^2=20.638+2.961$

$QR^2=23.59$

$QR=sqrt 23.59$

$QR=4.58cm$

NOTE:

This may look long winded but it is incredibly logical based on the knowledge you should already have on Pythagoras and trigonometry.

Example 2

In the diagram below find $theta$

Find the angle to the following diagram

Redraw

Redraw the diagram

First find $x$ for triangle I

First find X of the triangle

Using Pythagoras’s theorem:

$4^2=(6.9-x)^2+Y^2$

$4^2=6.9^2-6.9x-6.9x+x^2+Y^2$

$16=47.61-13.8x+x^2+Y^2$

$16-47.61+13.8x-x^2=Y^2$

$-31.61+13.8x-x^2=Y^2$ …………… Equation 1

For triangle II

Find y of the second triangle

$Y^2+x^2=4.2^2$

Therefore $Y^2=4.2^2-x^2$ ……………. Equation 2

Substitute equation 2 into equation 1

$-31.61+13.8x-x^2=4.2^2-x^2$

$-31.61+13.8x=4.2^2-x^2+x^2$

$-31.61+13.8x=4.2^2$

$13.8x=4.2^2+31.61$

$x=(4.2^2+31.61)/13.8$

Answer:

$x=3.56cm$ long

Now use cosine on triangle II

Now use cosine on triangle 2 to get an answer

Find $theta$

$costheta=3.56/4.2$

$costheta=0.849$

$theta=31.8^@$

Again this may look long winded but the example can be completed without the need of using the cosine rule, it just takes practice.