Right triangle definitions of trig functions and related applications

Some of the most fundamental trigonometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle:

right triangle

At this stage in the course the Pythagorean Theorem should be second nature to you. If you have not yet learned it, learn it now.

$a^{2}+b^{2}=c^{2}$

$a$ and $b$ of course are the legs or the adjacent and opposite edges, and $c$ is the hypotenuse, the longest side, the side that does not include the right angle. This formula only works for a right angle triangle. If the angle shown as a right angle in the diagram were obtuse, larger than a right angle, then $c^{2}$ would be larger than the Pythagorean sum. If the angle shown as a right angle were smaller than a right angle, then $c^{2}$ would be smaller than $a^{2}+b^{2}$ .

Sine, Cosine, and Tangent for Right Triangles

Sine, Cosine, and Tangent are all functions of an angle, which are useful in right triangle calculations. For an angle designated as θ, the sine function is abbreviated as sin θ, the cosine function is abbreviated as cos θ, and the tangent function is abbreviated as tan θ. For any
angle θ, sin θ, cos θ, and tan θ are each single determined values and if θ is a known value, sin θ, cos θ, and tan θ can be looked up in a table or found with a calculator. There is a table listing these function values at the end of this section. For an angle between listed values, the sine, cosine, or tangent of that angle can be estimated from the values in the table. Conversely, if a number is known to be the sine, cosine, or tangent of an angle, then such tables could be used in reverse to find (or estimate) the value of a corresponding angle.

These three functions are related to right triangles in the following ways:

In a right triangle,

• the sine of a non-right angle equals the length of the leg opposite that angle divided by the length of the hypotenuse.

• the cosine of a non-right angle equals the length of the leg adjacent to it divided by the length of the hypotenuse.

• the tangent of a non-right angle equals the length of the leg opposite that angle divided by the length of the leg adjacent to it.

For any value of θ where cos θ ≠ 0,

$\qquad \tan \theta ={\frac {\sin \theta }{\cos \theta }}$.

If one considers the diagram representing a right triangle with the two non-right angles θ₁and θ₂, and the side lengths a,b,c as shown here:

For the functions of angle θ₁:

$\sin \theta _{1}={\frac {b}{c}}\qquad \cos \theta _{1}={\frac {a}{c}}\qquad \tan \theta _{1}={\frac {b}{a}}$

Analogously, for the functions of angle θ₂:

$\sin \theta _{2}={\frac {a}{c}}\qquad \cos \theta _{2}={\frac {b}{c}}\qquad \tan \theta _{2}={\frac {a}{b}}$

Resources

• Right triangle definitions of trig functions and related applications. Written notes created by Professor Esparza, UTSA.