In real life situations involving right-angled triangles, we can use trigonometry to solve problem we normally couldn't by finding angles and lengths that are otherwise impossible to calculate.
To find missing angles in right-angled triangles using trigonometry, we require at least two known sides and the trigonometric ratios.
Remember that the trigonometric ratios for a right-angled triangle are:\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}
To isolate the angle in each of these relationships, we can apply the inverse trigonometric function to each side of the equation. This will give us: \theta =\sin^{-1}\left(\dfrac{\text{Opposite }}{\text{Hypotenuse }}\right) \quad \quad \theta =\cos^{-1}\left(\dfrac{\text{Adjacent }}{\text{Hypotenuse }}\right) \quad \quad \theta =\tan^{-1}\left(\dfrac{\text{Opposite }}{\text{Adjacent }}\right)
Any of these relationships can be used to find \theta depending on which side lengths of the triangle are known.
While we may represent the inverse trigonometric functions using an index of -1, they are not reciprocals of the original functions.
For example: \sin^{-1} \theta \neq \dfrac{1}{\sin \theta}
To find missing sides is right-angled triangles using trigonometry, we require at least one angle (other than the right angle) and at least one other side length.
We can find a missing side length by expressing it in a trigonometric ratio as the only unknown value, and then solve the equation to find that value.
Depending on the given angle, given side and missing side, we will need to use one of the three trigonometric ratios so that all relevant information is in the equation.
AB is a tangent to a circle with centre O. OB is 20 cm long and cuts the circle at C.
Find the length of BC to two decimal places.
To find missing angles in right-angled triangles using trigonometry, we require at least two known sides.
To find missing sides in right-angled triangles using trigonometry, we require at least one angle (other than the right angle) and at least one other side length.
In some cases, a single application of trigonometry on a single right-angled triangle will not be enough to find the values we are looking for.
In these cases, we can use trigonometry once to find a new value, and then use trigonometry again with our new value to find another new value. We can repeat this process as many times as is necessary to find the value that we are looking for.
Find the size of angle z. Give the answer in degrees, minutes and seconds, rounding to the nearest second.
We can use trigonometry once to find a new value, and then use trigonometry again with our new value to find another new value. We can repeat this process as many times as is necessary to find the value that we are looking for.
In other cases, multiple applications of trigonometry need to be used at the same time in order to solve for a common factor.
These cases arise when multiple trigonometric ratios involve the same missing value. If some relationship between these ratios is known then the missing value can be solved for.
Find the length of AB to two decimal places.
In trigonometry, there are certain cases where multiple trigonometric ratios involve the same missing value. We can find the missing value if some relationship between these ratios is known.