Trigonometry
A ratio is a way of stating a mathematical relationship comparing two quantities and is often represented as a fraction. In a right-angled triangle the ratios of the sides are called trigonometric ratios.
The three basic trigonometric ratios are Sine, Cosine and Tangent. These names are often shortened to become sin, cos and tan respectively. They are given by the ratio of sides relative to an angle, the reference angle.
Trigonometric ratios can be used to find an unknown side-length or an unknown interior angle in a right-angled triangle.
Labelling sides of a right-angled triangle
The hypotenuse is the longest length in the triangle, which is always opposite the $90^\circ$90° angle.
The opposite side-length is opposite to the reference angle $\theta$θ .
The adjacent side-length is adjacent to the reference angle $\theta$θ.
$\sin\theta$sinθ = $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse = $\frac{O}{H}$OH
$\cos\theta$cosθ = $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{A}{H}$AH
$\tan\theta$tanθ = $\frac{Opposite}{Adjacent}$OppositeAdjacent = $\frac{O}{A}$OA
The mnemonic of SOH CAH TOA is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.
Practice question
Question 1
Consider the triangle in the figure. If $\sin\theta=\frac{4}{5}$sinθ=45:
Which angle is represented by $\theta$θ?
$\angle BAC$∠BAC
A$\angle BCA$∠BCA
B$\angle ABC$∠ABC
CFind the value of $\cos\theta$cosθ. Express your answer as a simplified fraction.
Find the value of $\tan\theta$tanθ. Express your answer as a simplified fraction.
Finding an unknown side-length
For a right-angled triangle, given an angle and a side length, it is possible to use trigonometric ratios to find the length of an unknown side.
Using the reference angle, label the sides opposite, adjacent and hypotenuse. Choose which ratio to use and solve the equation for the unknown.
Practice question
Question 2
Find the value of $f$f, correct to two decimal places.
Finding an unknown angle
Inverse trigonometric ratios are used to find unknown interior angles in right-angled triangles, providing at least two of the triangle's side-lengths are known.
Consider the following right-angled triangle with unknown interior angle $\theta$θ
$\theta=\sin^{-1}\left(\frac{O}{H}\right)$θ=sin−1(OH)
$\theta=\cos^{-1}\left(\frac{A}{H}\right)$θ=cos−1(AH)
$\theta=\tan^{-1}\left(\frac{O}{A}\right)$θ=tan−1(OA)
Practice questions
Question 4
If $\cos\theta=0.256$cosθ=0.256
Find $\theta$θ, writing your answer to the nearest degree.
Question 5
Find the value of $x$x to the nearest degree.
A right-angled triangle with vertices labeled A, B and C. Vertex A is at the top, B at the bottom right, and C at the bottom left. A small square at vertex A indicates that it is a right angle. Side interval(BC), which is the side opposite vertex A, is the hypotenuse and is marked with a length of 25. The angle located at vertex B is labelled x. Side interval(AB), descending from the right angle at vertex A to vertex B, is marked with a length of 7, and is adjacent to the angle x. Side interval(AC) is opposite the angle x.
Question 6
Consider the given figure.
Find the unknown angle $x$x, correct to two decimal places.
Find $y$y, correct to two decimal places.
Find $z$z correct to two decimal places.