Right Triangle Trigonometry Formulas
In trigonometry, there are six functions that are used to determine the relationships between sides and angles of right triangles. The "basic" functions are sine, cosine, and tangent. These operations are only performed for right triangles. An easy way to remember each and its side-length ratio is S&CAO. This means that sin is equal to the side length of the opposite side over the hypotenuse, where the opposite side is the side opposite of the angle, and the hypotenuse is the longest side and the side opposite the right angle. The next ratio is cos, which is the adjacent side over the hypotenuse. The adjacent side is the side next to the angle, but not the hypotenuse. The third ratio is tangent, where the ratio is opposite over adjacent. The other three functions are the inverses of the aforementioned ratios, only with different names. They are Cosecant (inverse of sine) 1/a, secant (inverse of cosine) H/A, and cotangent (inverse of tangent) 1/tan. These are not to be confused with sin, cos, and tan on calculators.
Solving Right Triangles
Solving a right triangle means finding all side lengths and angle measures of a right triangle. For example, if we have an angle of 53° in a right triangle, we can use the Pythagorean theorem to find the length of the side a, which would be equal to √36.
Special Right Triangles
Special right triangles are triangles with certain angle measures, resulting in specific side relationships. There are two special right triangles: 30-60-90 and 45-45-90. The 45-45-90 triangles have one 90° angle and two 45° angles, making them isosceles. The relationship of any leg to the hypotenuse is x√2. In a unit circle, side lengths are √²/2, √²/2, and 1. The 30-60-90 triangles have one 90° angle, one 30⁰ angle, and one 60⁰ angle, where the side opposite the 30⁰ is the short leg and the side opposite 60⁰ is the long leg. The hypotenuse is double the length of the short leg, and the length of the long leg is x√3 of the short leg.
Unit Circle and Co-terminal Angles
A radian is a unit of angle measure, similar to degrees. On the unit circle, there are benchmark angles in the four quadrants, 0⁰, 90⁰, 180⁰, and 270⁰. These angles, along with any other angle, may be converted to radians. The formula for degrees to radians is also applicable for any angle measure. The benchmark angles help in quickly converting degrees to radians and vice versa. There are also other common angles to know, such as 45° and 30°.
The unit circle can be used to evaluate all six trig functions of any angle without using a calculator. This is because, on the unit circle, sin = y, cos = x, tan = y/x, and the reciprocals of each. Other common angles to know are 60°, 90°, and 270°. Radians can be converted to degrees by using logic, and understanding the relationships between degrees and radians.
Positive and Negative Trig Functions
Trigonometric functions can be negative or positive because they are on the coordinate plane. An easy way to remember which functions are positive or negative for each quadrant is to remember the acronym "All Students Take Calculus." The first letter stands for "All," meaning all functions are positive. "C" is for "cosine," so only cosine is positive in the second quadrant, and so on.
For more detailed information, you can access a right triangle trigonometry worksheet with answers in PDF format, right triangle trigonometry notes, as well as a solving right triangles worksheet. There are also right triangle trigonometry calculators and special right triangles calculators available for further practice and understanding.
