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For most angles $\theta $ , it is very difficult to calculate the values of $sin\theta $ , $cos\theta $ and $tan\theta $ . One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorum. The square of the hypothenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.
${0}^{\circ}$ | ${30}^{\circ}$ | ${45}^{\circ}$ | ${60}^{\circ}$ | ${90}^{\circ}$ | ${180}^{\circ}$ | |
$cos\theta $ | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | 0 | $-1$ |
$sin\theta $ | 0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | 1 | 0 |
$tan\theta $ | 0 | $\frac{1}{\sqrt{3}}$ | 1 | $\sqrt{3}$ | $-$ | 0 |
These values are useful when asked to solve a problem involving trig functions without using a calculator.
Find the length of x in the following triangle.
In this case you have an angle ( ${50}^{\circ}$ ), the opposite side and the hypotenuse.
So you should use $sin$
Use the sin button on your calculator
Find the value of $\theta $ in the following triangle.
In this case you have the opposite side and the hypotenuse to the angle $\theta $ .
So you should use $tan$
Since you are finding the angle ,
use ${tan}^{-1}$ on your calculator
Don't forget to set your calculator to `deg' mode!
The following videos provide a summary of what you have learnt so far.
Find the length of the sides marked with letters. Give answers correct to 2 decimal places.
Trigonometry was probably invented in ancient civilisations to solve practical problems such as building construction and navigating by the stars. In this section we will show how trigonometry can be used to solve some other practical problems.
One simple task is to find the height of a building by using trigonometry. We could just use a tape measure lowered from the roof, but this is impractical (and dangerous) for tall buildings. It is much more sensible to measure a distance along the ground and use trigonometry to find the height of the building.
[link] shows a building whose height we do not know. We have walked 100 m away from the building and measured the angle from the ground up to the top of the building. This angle is found to be $38,{7}^{\circ}$ . We call this angle the angle of elevation . As you can see from [link] , we now have a right-angled triangle. As we know the length of one side and an angle, we can calculate the height of the triangle, which is the height of the building we are trying to find.
If we examine the figure, we see that we have the opposite and the adjacent of the angle of elevation and we can write:
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