How does the unit circle allow the trigonometric functions to be defined for all real numbers instead of just for acute angles?

1 answer:

8 0

Every point in the unit circle is identified either by its coordinates $(x,y)$ or by the angle it forms with the x-axis, $\alpha$ .

The trigonometric functions associate with every angle $\alpha$ and the correspondant $(x,y)$ coordinates the two values

$\cos(\alpha)=x,\quad\sin(\alpha)=y$

This procedure can be done for every angle $\alpha$ , so you don't have to work with acute angles only.

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zavuch27 [327]

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stepladder [879]

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$\displaystyle 64$

General Formulas and Concepts:

Calculus

Limits

Limit Rule [Variable Direct Substitution]: $\displaystyle \lim_{x \to c} x = c$

Limit Rule [Variable Direct Substitution Exponential]: $\displaystyle \lim_{x \to c} x^n = c^n$

Limit Property [Multiplied Constant]: $\displaystyle \lim_{x \to c} bf(x) = b \lim_{x \to c} f(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \lim_{x \to 0} f(x) = 4$

Step 2: Solve

Rewrite [Limit Property - Multiplied Constant]: $\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = \frac{1}{4} \lim_{x \to 0} [f(x)]^4$
Evaluate limit [Limit Rule - Variable Direct Substitution Exponential]: $\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = \frac{1}{4}(4^4)$
Simplify: $\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = 64$