Sine, cosine and tangent
Grow and shrink a right triangle and watch its side ratios stay fixed: they depend only on the angle.
Standards: G-SRT.C.6G-SRT.C.8
1Wonder
How tall is that tree? You can’t climb it with a tape measure. But if you walk 20 steps away and measure the angle up to the top, a right triangle will tell you the height. The secret is that the angle alone fixes the shape of the triangle.
2Explore
This activity needs a bigger screen
Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.
- Keep the angle at 35° and drag Size. Every side changes. What about sin, cos and tan?
- Make the angle bigger. Which ratio grows? Which one shrinks?
- At 45°, what is tan θ? Why do the opposite and adjacent sides match?
- Find the angle where sin θ is exactly 0.5.
3Explain
Name the sides from the angle
Stand at the angle θ. The side across from you is the opposite. The side next to you (not the longest) is the adjacent. The longest side, across from the right angle, is the hypotenuse.
In a right triangle, the angle alone fixes the ratios of the sides. sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. Know one side and one angle, and you can find the other sides.
4Practice
In a right triangle, the side across from the angle θ is called the…
5Recap
- Name the sides from θ: opposite, adjacent, hypotenuse.
- Same angle → same shape → same ratios, whatever the size.
- SOH · CAH · TOA: sin = O ÷ H, cos = A ÷ H, tan = O ÷ A.
- Use them to find lengths you can’t measure directly.