Lesson 46: Finding Trigonometric Functions and their Reciprocals

New Concept

A function whose rule is a trigonometric ratio is a trigonometric function. For example, the sine of angle $A$ is:

Why it matters

Trigonometry extends algebra from abstract equations to the physical world, allowing us to describe the fixed relationships within geometric shapes. Mastering these ratios is your first step toward modeling everything from sound waves and light to engineering and astronomy.

What’s next

Next, you’ll master the core trigonometric functions—sine, cosine, and tangent—and their reciprocals to solve for unknowns in right triangles.

The sine, cosine, and tangent of angle $A$ are defined as:

For a right triangle with side opposite $\angle A$ as 5, adjacent as 12, and hypotenuse 13: $\sin A = \frac{5}{13}$.
For the same triangle, the cosine is: $\cos A = \frac{12}{13}$.
And the tangent for that same angle is: $\tan A = \frac{5}{12}$.

Just remember the secret code SOH-CAH-TOA! These are special ratios that connect an angle's measure to the lengths of a right triangle's sides. They're the key to solving for missing pieces.

The cosecant, secant, and cotangent are reciprocals of the sine, cosine, and tangent functions:

If $\sin A = \frac{8}{17}$, then its reciprocal is $\csc A = \frac{17}{8}$.
If $\cos A = \frac{15}{17}$, then its reciprocal is $\sec A = \frac{17}{15}$.
If $\tan A = \frac{8}{15}$, then its reciprocal is $\cot A = \frac{15}{8}$.

These three are just the main trig functions flipped upside down! If you know SOH-CAH-TOA, you know these. Just flip the fraction of sine, cosine, or tangent to find them.

The side adjacent to an angle is one of the sides that forms the angle but is not the hypotenuse.

In a right triangle with vertices $P, Q, R$ (right angle at $Q$), the side adjacent to $\angle P$ is $PQ$.
In the same triangle $PQR$, the side adjacent to $\angle R$ is $QR$.
The hypotenuse $PR$ is never considered the adjacent side for acute angles $\angle P$ or $\angle R$.

Adjacent means 'next to.' In a right triangle, an angle is formed by two sides. The one that is not the hypotenuse is the adjacent side. It's the angle's non-slanted buddy.