which of the following represents the cosine of 70 degrees
The Correct Answer and Explanation is:
The cosine of 70 degrees, denoted as cos70∘\cos 70^\circcos70∘, represents the ratio of the adjacent side to the hypotenuse in a right triangle. It can be expressed as:cos70∘≈0.3420\cos 70^\circ \approx 0.3420cos70∘≈0.3420
Explanation:
To understand why cos70∘≈0.3420\cos 70^\circ \approx 0.3420cos70∘≈0.3420, we can recall that cosine is one of the trigonometric functions, which relates the angles of a right triangle to the lengths of its sides. Specifically:
- The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
- Mathematically, for an angle θ\thetaθ in a right triangle: cos(θ)=Adjacent sideHypotenuse\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}cos(θ)=HypotenuseAdjacent side
For a specific angle, 70∘70^\circ70∘, this ratio is constant. If you use a calculator or a trigonometric table to find cos70∘\cos 70^\circcos70∘, it approximates to 0.3420. This value is crucial in various applications, especially in fields like physics, engineering, and computer graphics, where precise angle measurements are necessary.
In practical terms, the cosine function for acute angles (between 0° and 90°) decreases as the angle increases. Since cos0∘=1\cos 0^\circ = 1cos0∘=1 and cos90∘=0\cos 90^\circ = 0cos90∘=0, we know that as the angle increases from 0° to 90°, the cosine value decreases gradually. For 70∘70^\circ70∘, it is roughly 0.3420, which reflects its position on this decreasing curve.
This approximation is commonly used in calculations involving angles, vectors, and waveforms, where precision is essential.
