Lesson Guide
Explanation
A right-angled triangle has three sides, and in trigonometry each one gets a name. Always start with the hypotenuse: it is the longest side, directly across from the right angle. Here that is A B. Next, look at the angle you care about, called theta, at A. The side directly across from theta, that does not touch it, is the opposite side, B C. The last side, A C, touches theta and is not the hypotenuse, so it is the adjacent side. Adjacent means next to.
Your Turn
In triangle PQR the right angle is at Q and θ is at R. Name the opposite side. Check: PQ, the side that does not touch R.
Watch Out
The hypotenuse also touches θ, but it is never called adjacent. Find the hypotenuse first, then the adjacent side is the other one touching θ.
Explanation
The words opposite and adjacent only make sense once you say which angle you are standing at. On the left you stand at angle A: B C is opposite and A C is adjacent. On the right you stand at angle B instead. Now A C is across from you, so it is the opposite side, and B C touches you, so it is the adjacent side. The hypotenuse is the same from both angles, because it is fixed by the right angle. So every time the angle changes, label the triangle again.
Your Turn
In triangle XYZ the right angle is at Z. Which side is adjacent to X, and which is adjacent to Y? Check: XZ is adjacent to X; YZ is adjacent to Y.
Watch Out
Do not keep the labels from a previous part. A side that was opposite A is adjacent to B.
Explanation
Once the sides are named, there are three ratios, and the word S O H C A H T O A remembers them. S O H: sine of theta is the opposite side over the hypotenuse. C A H: cosine is the adjacent side over the hypotenuse. T O A: tangent is the opposite side over the adjacent side. In this three, four, five triangle, from theta the opposite side is three, the adjacent side is four and the hypotenuse is five. So sine theta is three fifths, cosine theta is four fifths, and tangent theta is three quarters.
Your Turn
From the angle at B in the same triangle, write sin B. Check: opposite B is AC = 4, so sin B = 4/5.
Watch Out
tan θ is opposite over adjacent. Adjacent over opposite gives 4/3, which is the tangent of the OTHER angle.
Explanation
Why can sine theta be called a property of the angle? Look at these two triangles. They have the same angle theta, so they are similar: the large one is the small one enlarged by a factor of two. Every side doubles, so the opposite side over the hypotenuse is three over five in the small triangle and six over ten in the large one. Both are zero point six. The size of the triangle cancels out. Any right-angled triangle with this angle theta has the same sine, the same cosine and the same tangent.
Your Turn
A third triangle with the same θ has hypotenuse 25. How long is its opposite side? Check: sin θ = 0.6, so it is 0.6 × 25 = 15.
Watch Out
Making the triangle bigger does not make sin θ bigger. Only changing the angle changes the ratio.
Explanation
A right-angled triangle has two acute angles, here alpha at A and beta at B, and they add to ninety degrees. From alpha, the side of three is opposite. From beta, that same side of three is adjacent. So sine alpha and cosine beta are both three fifths. In the same way cosine alpha and sine beta are both four fifths. The tangents flip: tangent alpha is three quarters and tangent beta is four thirds. Once you know one angle's ratios, you know the other angle's for free. Quiz and exercises now.
Your Turn
If cos β = 4/5 in this triangle, what is sin α? Check: the side adjacent to β is opposite α, so sin α = 4/5.
Watch Out
sin α = cos β only works when α and β are the two acute angles of the same right-angled triangle.
📌 Key Points — this lesson in a nutshell
- Name the sides from the angle θ: the hypotenuse is opposite the right angle, the opposite side is across from θ, and the adjacent side is next to θ (and is not the hypotenuse).
- Change the angle and the opposite and adjacent sides swap. The hypotenuse never changes.
- SOH CAH TOA: sin θ = \(\dfrac{O}{H}\), cos θ = \(\dfrac{A}{H}\), tan θ = \(\dfrac{O}{A}\).
- Triangles with the same angle are similar, so the ratios depend only on the angle, never on the size of the triangle.
- If the acute angles are α and β, then sin α = cos β, cos α = sin β, and tan β = 1 ÷ tan α.
Step 1: Introduction
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