Did you know that trigonometry can be used to calculate the position from the shore to a ship?

The person on the shore can find the angle between the shore and the ship and the ship's captain can find the angle from the ship to shore. It looks a bit like a pirate ship, so best we can locate their exact position in case we need help!

The **sine rule** is used to **calculate angles** in a **non-right angled triangle.** The formula is:

It looks the same as the sine rule for finding a side length. The difference is that the terms have been turned upside down with the angles (sin) being on the top.

You will need a **scientific calculator** in order to get the sin^{-1} button. Make sure it is set to degrees.

__Example 1__

Calculate angle x.

__Step 1:__**Label** **the angles** of the triangle if it hasn't already been labelled. Our angles are A, B, and C. You are free to pick the order they go in.

__Step 2:__**Label the lengths**. This is where you have to be *very* precise, otherwise the formula will not work. The length **opposite** angle **A** __must__ be called **a**, the length **opposite** angle **B** __must__ be called **b**, and the length opposite angle **C** __must__ be called **c.**

A = 60°

B = x

C = (blank)

a = 7cm

b = 6cm

c = (blank)

__Step 3:__**Write the formula **that you will need. Notice that length c and angle C are both blank, so in this case, we can leave them out:

**sin A / a = sin B / b**

** Step 4: Substitute **your known values:

sin 60 / 7 = sin x / 6

** Step 5: Rearrange **the formula to make

**sin x**the subject:

**sin x** **=** 6 x (sin 60 / 7)

**sin x = ** 0.74 ...

** Step 6:** Use inverse sin ("

*) to find the angle. (Just remember to press "shift" + "sin" on your calculator to make it appear).*

**sin**"^{-1}

**x = sin ^{-1 }(0.74...)**

**x = 47.9**° (to 3 significant figures)

** NOTE:** If you try to put it all into the calculator at once, the rules of BIDMAS aren't followed and your answer will be incorrect.

Let's give it a try...