91Ó°ÊÓ

The sine function is a crucial part of trigonometry, dealing with the relationship between the angles and lengths of right triangles. In a right triangle, the sine of an angle (usually denoted by \( \sin \theta \) ) is defined as the ratio of the length of the side opposite to the angle to the hypotenuse's length.
This can be remembered with the mnemonic SOH-CAH-TOA, where for sine, SOH stands for "Sine equals Opposite over Hypotenuse."
The sine function is periodic, meaning it repeats its values in a regular interval, specifically every \( 360^{\circ} \) or \( 2\pi \) radians.

• The graph of sine is a smooth wave beginning at 0, peaking at 1, and dipping to -1, before returning to 0.
• It is an odd function, meaning that \( \sin(-\theta) = -\sin(\theta) \).

This property helps in solving larger trigonometry problems by breaking them into simpler parts, like how we find \( \sin 75^{\circ} \) using smaller angles.

The Sum Formula for Sine is a powerful identity in trigonometry that allows us to find the sine of the sum of two angles. It is given by:\[ \sin(a+b) = \sin a \cos b + \cos a \sin b \]This formula is particularly useful when the angles involved are not standard angles for which exact values are commonly known.
By expressing a complex angle as a sum of simpler, well-known angles, the formula lets us find the exact sine value efficiently.
For example, to find \( \sin 75^{\circ} \), recognize that this angle is the sum of \( 45^{\circ} \) and \( 30^{\circ} \).

• Using the formula, \( \sin 75^{\circ} = \sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \).

This concept is a great example of how breaking down problems can simplify your calculations.

Exact values in trigonometry refer to the specific values of sine, cosine, and tangent for commonly used angles like \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \). These values are often expressed in terms of square roots or fractions, and are derived from the properties of 30-60-90 and 45-45-90 triangles.
Knowing these exact values allows for precise calculations without the need for a calculator or approximation.

• \( \sin 30^{\circ} = \frac{1}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), \( \cos 60^{\circ} = \frac{1}{2} \)

By memorizing these values, you can efficiently solve problems involving angles expressed as sums or multiples of these familiar angles, such as in finding \( \sin 75^{\circ} \), which utilizes known values for \( 45^{\circ} \) and \( 30^{\circ} \).