Key takeaway: The cosine of an acute angle in a right triangle is the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse.
Grade 10 – GeometryG-SRT.C.6Trigonometry
Definition of Cosine
The cosine of an angle compares the leg that touches the angle with the hypotenuse. Like all trigonometric ratios, it has no units, and it depends only on the angle, since triangles with equal angles are similar.
A taller angle means a shorter touching leg, so as an acute angle grows, its cosine shrinks from nearly 1 toward 0. Cosine is the natural tool for horizontal distances: when you know a slanted length and its angle with the ground, the cosine gives the ground distance covered.
Formula and notation
Written \(\cos\theta\), and read “cosine of theta”:
\[ \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
To find an angle from a known ratio, use \(\theta = \cos^{-1}(\text{ratio})\).
Key facts
- It is the CAH in SOH-CAH-TOA: cosine is adjacent over hypotenuse.
- Acute-angle values lie between 0 and 1, and they fall as the angle grows.
- It equals the sine of the complementary angle: cos 36 degrees equals sin 54 degrees.
- The adjacent leg is the one touching the angle, never the hypotenuse.
Cosine example
A kite string 60 ft long makes a \(54^{\circ}\) angle with level ground. How far along the ground is the kite from the person holding it? Use \(\cos 54^{\circ} \approx 0.5878\). Then find the angle if the ground distance were 48 ft with the same string.
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The ground distance \(d\) is adjacent and the string is the hypotenuse: \(\cos 54^{\circ} = \frac{d}{60}\).
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Solve: \(d = 60 \times 0.5878 \approx 35.3\) ft.
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For 48 ft: \(\cos\theta = \frac{48}{60} = 0.8\), so \(\theta = \cos^{-1}(0.8) \approx 36.9^{\circ}\).
The kite is about 35.3 ft away along the ground; a 48-ft distance would mean an angle of about \(36.9^{\circ}\).
Common mistake
Watch out
Ravi finds the ground distance by dividing: \(d = \frac{60}{0.5878} \approx 102\) ft, longer than the string itself.
Why it is wrong
He solved the equation backward. Since \(\cos 54^{\circ} = \frac{d}{60}\), multiply both sides by 60 to get \(d = 60 \times 0.5878\). A leg can never be longer than the hypotenuse, so an answer of 102 ft should set off an alarm.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Trigonometry