When working in a right angled triangle, the longest side is known as the hypotenuse. The other two sides are known as the opposite and the adjacent. The adjacent is the side next to a marked angle, and the opposite side is opposite this angle.
For a right angled triangle, the sine, cosine and tangent of the angle θ are defined as:
Sin θ will always have the same value for any particular angle, regardless of the size of the triangle. The same is true for cosθ and tanθ.
Worked Examples
For the triangle and angle shown, state which side is:
the hypotenuse
Show meThe hypotenuse is the longest side, which for this triangle is CB.
the adjacent
Show meThe adjacent is the side that is next to the angle θ , which for this triangle is AB.
the opposite.
Show meThe opposite side is the side that is opposite the angle θ , which for this triangle is AC.
Write down the values of sinθ , cosθ and tanθ for the triangle shown. Then use a calculator to find the angle in each case.
| opposite | = 8 |
| adjacent | = 6 |
| hypotenuse | = 10 |
| sinθ | = |
| = | |
| = 0.8 |
| cosθ | = |
| = | |
| = 0.6 |
| tanθ | = |
| = | |
| = |
Using a calculator gives θ = 53.1° (correct to 1 decimal place) in each case.
Note
In the triangle opposite, we know that
sinθ = , cosθ =
Thus
| sin2θ + cos2θ | = ⎛ ⎝⎞ ⎠2 + ⎛ ⎝⎞ ⎠2 |
| = + | |
| = |
But by Pythagoras' Theorem, we also know that
a2 = b2 + c2
Hence
sin2θ + cos2θ = = 1
This result
is a useful result. For example, when θ = 45°,
sin 45 = cos 45 (as the two sides c and b are equal)
and hence,
| sin2 45 + sin2 45 = 1 | |||
| 2sin2 45 = 1 | |||
| sin2 45 = | |||
| or | sin 45 = cos 45 | = | (You can check this on your calculator.) |
Exercises
