Right Triangle Trigonometry

Right Triangle Trigonometry 

When we look at a right triangle,we can see that the shape is made up of three sides. These three sides each have their own distinct name, which vary depending on the location of the angle in question (We will use Theta, written as "θ"). The identity of the sides can be seen in the image below:

Where:

• H = Hypotenuse (Always opposite of the right angle)
• A = Side Adjacent to θ
• O = Side Opposite to θ

The identities of the three sides of a triangle can be used to define all of the six trigonometric functions (Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent). The equations used to find the trigonometric functions using the three sides of a right triangle are as follows:

• Sin θ = O/H
• Cos θ = A/H
• Tan θ = O/A
• Csc θ = H/O
• Sec θ = H/A
• Cot θ = A/O

In addition to these equations, there are many other ways one can relate the sides and angles of a right triangle. These relationships can be defined as Trigonometric Identities, and there are many different kinds of them. Trigonometric identities are unique in the way that they are always true, no matter what the input of the equation is, meaning that the Domain is equal to all real numbers. There are three main categories of trigonometric identities used to relate the six trigonometric functions to each other:

1). Reciprocal Identities - Shows the relationship between each trig function and its reciprocal

• Sin θ = 1/Csc θ
• Cos θ = 1/Sec θ
• Tan θ = 1/Cot θ
• Csc θ = 1/Sin θ
• Sec θ = 1/Cos θ
• Cot θ = 1/Tan θ

2). Quotient Identities - Shows the relationship between two trig functions and the quotient of them                                            divided together

• Tan θ = Sin θ/Cos θ
• Cot θ = Cos θ/Sin θ

3). Pythagorean Identities - Shows the relationship between trig functions through the Pythagorean                                                  Theorem

• (Sin θ)^2 + (Cos θ)^2 = 1
• 1 + (Tan θ)^2 = (Sec θ)^2 
• 1 + (Cot θ)^2 = (Csc θ)^2

*The Second and Third identities can be derived from the First one.

These identities can be used in numerous ways throughout Trigonometry and Pre-Calculus, as well as Calculus. They can be used to condense and expand expressions, and solve for trigonometric functions when needed. Trigonometric Identities are extremely useful when proving relationships between functions, as well as deriving new proofs. 

Here is an Example of a problem where Trig Identities can be used to prove an equation:

Prove: (Csc θ)(Tan θ) = Sec θ

One can use the reciprocal identity to transform both Cosecant and Tangent into terms of Sine and Cosine.

(1/Sin θ)(Sin θ/Cos θ) = Sec θ

Then, one can cancel out the Sin θ's and continue to simplify.

1/Cos θ = Sec θ

Now we see that both sides of the equation are equal due to the reciprocal identity.