Glossary
30-60-90 Triangle
A special right triangle with angles measuring 30°, 60°, and 90°, and specific side ratios of 1:√3:2.
Example:
Recognizing a 30-60-90 Triangle can save you time by allowing you to quickly determine side lengths without using trigonometric functions.
45-45-90 Triangle
A special right triangle with angles measuring 45°, 45°, and 90°, and specific side ratios of 1:1:√2.
Example:
Finding the diagonal of a square is a classic application of the 45-45-90 Triangle properties.
Adjacent side
In a right triangle, the side that is next to a given acute angle and is not the hypotenuse.
Example:
If you're looking at a building, your distance from its base along the ground is the Adjacent side to your angle of elevation.
Angle of Depression
The angle formed when an observer looks downward from a horizontal line to an object below.
Example:
From the top of a cliff, looking down at a boat on the water, you would measure an Angle of Depression.
Angle of Elevation
The angle formed when an observer looks upward from a horizontal line to an object above.
Example:
A pilot looking up at a control tower from the runway would measure an Angle of Elevation.
Cosecant
The reciprocal of the sine function, defined as the ratio of the hypotenuse to the opposite side.
Example:
While less common on the SAT, knowing Cosecant is 1/sine can help simplify expressions in more advanced problems.
Cosine
In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.
Example:
To determine how far a ladder's base is from a wall, given its length and the angle it makes with the ground, you would use Cosine.
Cotangent
The reciprocal of the tangent function, defined as the ratio of the adjacent side to the opposite side.
Example:
Sometimes, using Cotangent can be more direct if you're working with the adjacent and opposite sides in a specific context.
Hypotenuse
The longest side of a right-angled triangle, always located opposite the 90-degree angle.
Example:
When building a ramp, the length of the ramp itself represents the Hypotenuse of the right triangle formed.
Inverse Cosine (Arccosine)
A function used to find the measure of an angle when the ratio of the adjacent side to the hypotenuse is known.
Example:
To find the angle a ramp makes with the ground, given its length and horizontal distance, you would use Inverse Cosine.
Inverse Sine (Arcsine)
A function used to find the measure of an angle when the ratio of the opposite side to the hypotenuse is known.
Example:
If you know the sine of an angle is 0.5, you use Inverse Sine to find that the angle is 30 degrees.
Inverse Tangent (Arctangent)
A function used to find the measure of an angle when the ratio of the opposite side to the adjacent side is known.
Example:
If you know the height of a building and your distance from it, Inverse Tangent helps you calculate the angle of elevation to its top.
Legs (of a right triangle)
The two sides of a right-angled triangle that form the 90-degree angle.
Example:
In a right triangle, the Legs are the sides that are not the hypotenuse.
Opposite side
In a right triangle, the side that is directly across from a given acute angle.
Example:
When standing at the base of a tree, the tree's height is the Opposite side to your angle of elevation.
Pythagorean Theorem
A fundamental theorem stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²).
Example:
If you know two sides of a right triangle, the Pythagorean Theorem is your go-to for finding the length of the third side.
Right Triangle Trigonometry
A branch of mathematics focused on the relationships between the angles and sides of right-angled triangles.
Example:
Understanding Right Triangle Trigonometry is essential for calculating the height of a flagpole given its shadow and the sun's angle.
SOH-CAH-TOA
An acronym used to remember the definitions of the three basic trigonometric ratios: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent).
Example:
When solving for a missing side, remembering SOH-CAH-TOA helps you quickly choose the correct trigonometric function.
Secant
The reciprocal of the cosine function, defined as the ratio of the hypotenuse to the adjacent side.
Example:
If you've calculated cosine, finding Secant is as simple as flipping the fraction.
Sine
In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.
Example:
If you know the angle of elevation and the length of a ramp, you can use Sine to find the vertical height it reaches.
Tangent
In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the side adjacent to that angle.
Example:
A drone flying at a certain altitude can use Tangent to calculate its horizontal distance from a landmark if it knows the angle of depression.