Cosine
In short: Cosine compares the ground covered with the path traveled: for a corner of a right triangle, it is the leg touching that corner divided by the hypotenuse.
GeometryTrigonometryG-SRT.C.6
What is Cosine?
The cosine of a corner \(A\) measures how much of the slanted path lies along the flat. Divide the leg that touches \(A\) by the hypotenuse: \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\).
For corners under \(90^{\circ}\) the value lies between 0 and 1, and it shrinks as the corner opens: \(\cos 0^{\circ} = 1\) while \(\cos 90^{\circ} = 0\). It also equals the sine of the complementary corner, so \(\cos 60^{\circ} = \sin 30^{\circ} = 0.5\).
How to spot it
- A diagram shows a marked corner, the slanted side, and the flat leg that touches the corner.
- A story asks how far along the ground a ramp, guy wire or ladder reaches at a given tilt.
- The slanted side and a leg next to the corner are known, and the corner size is wanted.
- A calculator key labeled cos appears, or the quotient adjacent over hypotenuse is written.
The method
- Mark the corner, label the leg touching it adjacent and the longest side the hypotenuse.
- Write \(\cos\theta = \frac{\text{adj}}{\text{hyp}}\) and substitute the known values.
- For a leg, multiply: \(\text{adj} = \text{hyp} \times \cos\theta\). For a corner, press \(\cos^{-1}\).
- Confirm the answer is shorter than the hypotenuse and the calculator uses degree mode.
Cosine: solved example
Problem
A loading ramp 26 feet long makes a \(22^{\circ}\) angle with the pavement. How far along the pavement does the ramp stretch? Use \(\cos 22^{\circ} \approx 0.9272\).
Solution
The horizontal distance touches the \(22^{\circ}\) corner, so it is the adjacent leg; the ramp is the hypotenuse.
From \(\cos 22^{\circ} = \frac{d}{26}\): \(d = 26 \times 0.9272 \approx 24.1\). The ramp covers about 24.1 feet of pavement.
Try it
Question 1
The flat leg touching corner \(\theta\) is 20 and the slanted side is 29. Write \(\cos\theta\) as a fraction and as a decimal.
Show the answer
\(\cos\theta = \frac{20}{29} \approx 0.6897\).
Question 2
A corner has \(\cos\theta = 0.6\) and the slanted side is 35. How long is the leg touching the corner?
Show the answer
\(35 \times 0.6 = 21\).
Cosine vs. Sine
Both divide by the hypotenuse; the numerator is what differs. Cosine takes the leg that touches the corner, sine takes the leg across from it. At \(30^{\circ}\), the first gives about 0.866 and the second gives 0.5. In the memory phrase, the C of CAH goes with adjacent.
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