Sine
In short: Sine turns a corner of a right triangle into a fraction: the side facing that corner, divided by the slanted longest side, the hypotenuse.
GeometryTrigonometryG-SRT.C.6
What is Sine?
The sine of a corner \(A\) answers one question: how tall is the triangle compared with its slanted edge? Take the leg across from \(A\) and divide by the hypotenuse, written \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\).
The hypotenuse is the longest edge, so for corners under \(90^{\circ}\) the quotient lands between 0 and 1. Any two right triangles sharing that corner give the same quotient regardless of size. That is why a calculator can store one value per angle, for instance \(\sin 30^{\circ} = 0.5\).
How to spot it
- A diagram shows a marked corner, the slanted side, and the side straight across from the corner.
- A story gives the length of a kite string, ladder, ramp or zip line and asks how high it climbs.
- You know the across-from side and the slanted side, and need the size of the corner.
- A calculator key labeled sin is mentioned, or the quotient opposite over hypotenuse is written.
The method
- Mark the corner, then label the side facing it opposite and the longest side the hypotenuse.
- Write \(\sin\theta = \frac{\text{opp}}{\text{hyp}}\) and fill in the two known quantities.
- To find a side, multiply: \(\text{opp} = \text{hyp} \times \sin\theta\). To find an angle, press \(\sin^{-1}\).
- Check that the calculator is in degree mode and that the answer is shorter than the hypotenuse.
Sine: solved example
Problem
At a kite festival in Kansas, Mateo lets out 85 feet of string, pulled straight, and it rises \(32^{\circ}\) above level ground. How high is the kite? Use \(\sin 32^{\circ} \approx 0.5299\).
Solution
The height faces the \(32^{\circ}\) corner, so it is the opposite side, and the string is the hypotenuse.
From \(\sin 32^{\circ} = \frac{h}{85}\): \(h = 85 \times 0.5299 \approx 45.04\). The kite hovers about 45 feet above the ground.
Try it
Question 1
One slanted edge measures 41 and the side facing corner \(B\) measures 9. Find \(\sin B\) to four decimals.
Show the answer
\(\sin B = \frac{9}{41} \approx 0.2195\).
Question 2
A corner has \(\sin\theta = 0.28\) and the slanted edge is 25. How long is the side facing the corner?
Show the answer
\(25 \times 0.28 = 7\).
Sine vs. Cosine
Both divide by the hypotenuse; only the top changes. Sine uses the leg across from the corner, cosine uses the leg touching it. In the 8-15-17 figure with corner \(A\), the first gives \(\frac{8}{17}\) and the second gives \(\frac{15}{17}\). The two numbers trade places when you switch to the other small corner.
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