In a right triangle, the opposite side is the side that is directly across from a given angle. It is used in trigonometric ratios such as sine and tangent.
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The length of the opposite side is essential for calculating the sine ($\sin$) of an angle: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.
The opposite side is also crucial for finding the tangent ($\tan$) of an angle: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.
In the unit circle, the y-coordinate represents the length of the opposite side for angles in standard position.
For a 45°-45°-90° triangle, the lengths of both legs (opposite and adjacent sides) are equal.
The Pythagorean theorem can involve the opposite side: $a^2 + b^2 = c^2$, where $a$ or $b$ may be designated as the opposite.
Review Questions
How do you calculate sine using the length of the opposite side?
What is the relationship between tangent and the lengths of opposite and adjacent sides?
In a right triangle with a hypotenuse of length 10 and an angle θ whose sine value is 0.6, what is the length of the opposite side?
Related terms
Hypotenuse: The longest side in a right triangle, located directly across from the right angle.
Adjacent Side: The side next to a given angle in a right triangle, excluding the hypotenuse.
Unit Circle: A circle with radius one centered at the origin on a coordinate plane; used to define trigonometric functions.