Triangles & Trigonometry Vocabulary and Problems
Hypotenuse
the longest side of a right triangle, always across from the right angle
Converse of the Pythagorean Theorem
(Proves Right Triangles) If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Common Pythagorean Triples and Some of their Multiples
3, 4, 5 6, 8, 10 9, 12, 15 3x, 4x, 5x 5, 12, 13 10, 24, 26 15, 36, 29 8, 15, 17 16, 30, 34 24, 45, 51 8x, 15x, 17x 7, 24, 25 14, 48, 50 21, 72, 75 7x, 24x, 25x
Triangle
A closed figure with 3 sides and 3 angles
Trigonometric Ratio
A ratio of the lengths of two sides in a right triangle to the angle.
Standard Position
A right triangle whose hypotenuse is a radius of the circle of radius 1 with center at the origin of a coordinate plane with one leg on the x-axis and the other leg perpendicular to the x-axis.
Pythagorean Triple
A set of three positive integers that satisfy the equation a²+b²=c².
Isosceles triangle
A triangle with 2 congruent sides
Equiangular Triangle
A triangle with all angles equal (60 degrees)
Acute triangle
A triangle with all angles less then 90 degrees
Equilateral Triangle
A triangle with all sides congruent
Scalene triangle
A triangle with no congruent sides
Obtuse triangle
A triangle with one angle greater than 90 degrees
Right triangle
A triangle with one right (90 degree) angle
Cosine Ratio
A trigonometric ratio for acute angles that involves the length of the adjacent leg of the angle in a right triangle and its hypotenuse. Length of leg adjacent to the angle/length of the hypotenuse.
Cosine Ratio (CAH)
A trigonometric ratio for acute angles that involves the length of the adjacent leg of the angle in a right triangle and its hypotenuse. Length of leg adjacent to the angle/length of the hypotenuse.
Sine Ratio
A trigonometric ratio for acute angles that involves the length of the opposite leg of the angle in a right triangle and its hypotenuse. Length of leg opposite the angle /length of the hypotenuse.
Sine Ratio (SOH)
A trigonometric ratio for acute angles that involves the length of the opposite leg of the angle in a right triangle and its hypotenuse. Length of leg opposite the angle /length of the hypotenuse.
Tangent Ratio
A trigonometric ratio for acute angles that involves the lengths of the legs of a right triangle. Length of leg opposite the angle /length of the leg adjacent the angle
Tangent Ratio (TOA)
A trigonometric ratio for acute angles that involves the lengths of the legs of a right triangle. Length of leg opposite the angle /length of the leg adjacent the angle
Solving a Right Triangle
Finding all side lengths and angle measures of a right triangle. Pythagorean Theorem is used with 2 sides if looking for the third side, and Trig Ratios (SOHCAHTOA) are used to find an angle given two sides, or finding a side given an angle and side
Pythagorean Inequalities Theorem
For any ∆ABC, where c is the length of the longest side, the following statements are true: If a²+b²>c², then ∆ABC is acute. If a²+b²=c², then ∆ABC is right.(Converse) If a²+b²<c², then ∆ABC is obtuse.
30°-60°-90° Triangle
In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.
45°-45°-90° Triangle
In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.
Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a²+b²=c², where c is always the hypotenuse.
Sine and Cosine of Complementary Angles
Let A and B be complementary angles, then: sin A = cos (90°- A) = cos B sin B = cos (90°-B) = cos A cos A = sin (90°-A) = sin B cos B = sin (90°-B) = sin A
Vertex
The angle opposite the non congruent side of an isoceles triangle
Angle of Depression
The angle that a downward line of sight makes with a horizontal line.
Angle of Elevation
The angle that an upward line of sight makes with a horizontal line.
Base angles
The angles opposite the congruent sides of an isoceles triangle
Area of a Triangle
The area of any triangle is base times height divided by 2. The height must be perpendicular to the base!
Altitude of a Triangle
The perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side (Height of the Triangle)
Rationalizing the Denominator
The process of eliminating a radical from the denominator of a fraction
Ordering side/angles of a triangle
The shortest side is across from the shortest angle The longest side is across from the biggest angle.
Opposite Side
The side away from the reference angle
Adjacent Side
The side between the reference angle and the right angle
Triangle Inequality Theorem
The sum of any 2 sides of a triangle must be greater than the measure of the third side. The sum of the 2 smaller sides must be greater than the measure of the third side
Exterior Angle Theorem
The sum of the remote interior angles is equal to the exterior angle
Leg of a Right Triangle
The two sides of the triangle that are adjacent to the right angle
Angle Sum of triangle
There are 180 degrees inside every triangle
Range for the third side of a triangle
To find the possible measures of the third side of a triangle: 1. Subtract the sides you know 2. Add the sides you know 3. The possible side lenghts are between these 2 numbers
Law of Sines
Used to find angles and sides of non-right triangles. Is used with triangles where the given information is in the format ASA, AAS, and SSA, (SSA can create ambiguous cases with no or 2 solutions)
Law of Cosines
Used to solve a side opposite the angle given a triangle in the format SAS. If rewritten to solve for the Angle can solve an Angle in SSS format, be careful to always solve for the biggest angle to prevent ambiguous cases.
Inverse Trigonometric Ratios
You can use sin⁻¹, cos⁻¹ and tan⁻¹ to find the measure of an angle in a right triangle given its sides.
Unit Circle
a circle with a radius of 1, centered at the origin
Median of a Triangle
a segment from a vertex to the midpoint of the opposite side
Radical
root symbols
Reference Angle
the acute angle formed by the terminal side of an angle in standard position and the x-axis.
