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Sine and Cosine in Trigonometry

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Questions and Answers

How are sine and cosine functions denoted for an angle θ?

$\sin (\theta)$ and $\cos (\theta)$

$\sin \theta$ and $\cos \theta$ (correct)

$\sin \theta$ and $\cosine \theta$

$\theta \sin$ and $\theta \cos$

How can sine and cosine functions be extended to any real value?

In terms of the lengths of certain line segments in a unit circle (correct)

Through solving differential equations

By using only infinite series

By using only Indian astronomy principles

What is the definition of sine for an acute angle in the context of a right triangle?

The ratio of the length of the hypotenuse to that of the adjacent leg

The ratio of the length of the hypotenuse to that of the opposite side

The ratio of the length of the side opposite the angle to the length of the hypotenuse (correct)

The ratio of the length of the adjacent leg to that of the hypotenuse

What are sine and cosine commonly used to model?

<p>Periodic phenomena such as sound and light waves</p> Signup and view all the answers

How are sine and cosine written using functional notation?

<p>$\sin$ and $\cos$</p> Signup and view all the answers

Study Notes

Sine and Cosine Function Denotation

Sine and cosine functions are denoted as sin(θ) and cos(θ) for an angle θ.

Extension to Real Values

Sine and cosine functions can be extended to any real value using the unit circle, establishing periodicity with a period of 2π.
Both functions are defined for all real numbers, enabling them to model angles beyond those found in right triangles.

Definition of Sine for Acute Angles

In the context of a right triangle, sine for an acute angle (θ) is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
Mathematically, this can be expressed as: sin(θ) = opposite/hypotenuse.

Applications of Sine and Cosine Functions

Sine and cosine functions are commonly used to model periodic phenomena, such as sound waves and oscillations.
They are also employed in various fields such as engineering, physics, and signal processing to represent cycles and waveforms.

Functional Notation

Sine and cosine can be written in functional notation as f(θ) = sin(θ) and g(θ) = cos(θ), indicating their use as mathematical functions of the angle θ.

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Description

Test your knowledge of sine and cosine functions, and their definitions in the context of right triangles. Explore the ratios of side lengths and how they relate to the angles in trigonometry.