Trigonometric Functions
Define and evaluate sine, cosine, and tangent using right triangle ratios SOH-CAH-TOA: link angle measures to side length ratios for any right triangle in Grade 10 trigonometry.
Key Concepts
For a point $Q(x, y)$ on the terminal side of an angle $\theta$ in standard position, where $r = \sqrt{x^2 + y^2}$, the trigonometric functions are defined as: $$\sin \theta = \frac{y}{r} \quad \cos \theta = \frac{x}{r} \quad \tan \theta = \frac{y}{x}, x \neq 0$$.
For a point $Q$ at $( 3, 4)$: First, find the distance $r = \sqrt{( 3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. Then, use the definitions to find the trigonometric values: $\sin \theta = \frac{y}{r} = \frac{4}{5}$, $\cos \theta = \frac{x}{r} = \frac{3}{5}$, and $\tan \theta = \frac{y}{x} = \frac{4}{3}$.
Any point on an angle's terminal side can be the corner of a right triangle. Its coordinates $(x, y)$ give you the triangle's legs, and the distance from the origin, $r$, is the hypotenuse. With just these three numbers, you can instantly find the sine, cosine, and tangent of the angle, no matter how large it is!
Common Questions
What are the six trigonometric functions and how are they defined?
The six trig functions defined from a right triangle are sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), tangent (opposite/adjacent), and their reciprocals: cosecant, secant, and cotangent. SOH-CAH-TOA covers the three primary ratios.
How do you evaluate sin, cos, and tan for a given acute angle?
Label the sides relative to the angle: the opposite side faces it, the adjacent side is next to it, and the hypotenuse is always the longest side. Divide the appropriate sides. In a 3-4-5 triangle with a 37-degree angle, sin(37) is approximately 3/5 = 0.6.
Why do trig function values remain constant for a given angle regardless of triangle size?
All right triangles with the same acute angle are similar, so their sides are proportional. Because trig ratios compare two sides of the same triangle, the proportionality cancels size differences, leaving the ratio constant for any given angle measure.