Anasayfa/ AP Calculus/ Trigonometric Functions
Quizler Videolar

πŸ“ Trigonometric Functions

$\sin x$, $\cos x$, $\tan x$ β€” definitions, graphs, identities, and key values
🎯 In this section you will learn

πŸ”΅ Radian Measure

In calculus, we always use radians for angle measurement. One full revolution is $2\pi$ radians.

$$ 180^\circ = \pi \text{ radians} \quad \Rightarrow \quad 1 \text{ radian} \approx 57.3^\circ $$
DegreesRadiansDegreesRadians
$0^\circ$$0$$90^\circ$$\frac{\pi}{2}$
$30^\circ$$\frac{\pi}{6}$$180^\circ$$\pi$
$45^\circ$$\frac{\pi}{4}$$270^\circ$$\frac{3\pi}{2}$
$60^\circ$$\frac{\pi}{3}$$360^\circ$$2\pi$

🎯 Unit Circle Definitions

For an angle $\theta$ in standard position, let $(x, y)$ be the point where the terminal side intersects the unit circle $x^2 + y^2 = 1$. Then:

$$ \sin \theta = y \quad \cos \theta = x \quad \tan \theta = \frac{y}{x} \quad (x \neq 0) $$ $$ \csc \theta = \frac{1}{y} \quad \sec \theta = \frac{1}{x} \quad \cot \theta = \frac{x}{y} $$
πŸ’‘ Remember: SOH CAH TOA
In a right triangle: $\sin = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan = \frac{\text{opposite}}{\text{adjacent}}$.

πŸ“Š Exact Values (Common Angles)

$\theta$ (rad)$\sin \theta$$\cos \theta$$\tan \theta$
$0$$0$$1$$0$
$\frac{\pi}{6}$$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\frac{1}{\sqrt{3}}$
$\frac{\pi}{4}$$\frac{\sqrt{2}}{2}$$\frac{\sqrt{2}}{2}$$1$
$\frac{\pi}{3}$$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\sqrt{3}$
$\frac{\pi}{2}$$1$$0$undefined

πŸ“ˆ Graphs of Sine, Cosine, and Tangent

Sine Function: $f(x) = \sin x$

Cosine Function: $f(x) = \cos x$

Tangent Function: $f(x) = \tan x = \frac{\sin x}{\cos x}$

Example 1Evaluating Trig Functions

Find $\sin\left(\frac{5\pi}{6}\right)$, $\cos\left(\frac{5\pi}{6}\right)$, and $\tan\left(\frac{5\pi}{6}\right)$.

β‘ 
Reference angle
$\frac{5\pi}{6}$ is in Quadrant II. Reference angle: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.
β‘‘
Signs in Quadrant II
Sine positive, cosine negative, tangent negative.
β‘’
Values
$\sin = +\frac{1}{2}$, $\cos = -\frac{\sqrt{3}}{2}$, $\tan = -\frac{1}{\sqrt{3}}$.

πŸ”— Fundamental Trigonometric Identities

$$ \sin^2 x + \cos^2 x = 1 $$ $$ 1 + \tan^2 x = \sec^2 x $$ $$ 1 + \cot^2 x = \csc^2 x $$ $$ \sin(2x) = 2\sin x \cos x $$ $$ \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x $$ $$ \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y $$ $$ \cos(x \pm y) = \cos x \cos y \mp \sin x \sin y $$
Example 2Using Identities

Simplify $\sin^2 x \cos x + \cos^3 x$.

Solution: Factor $\cos x$: $\cos x (\sin^2 x + \cos^2 x) = \cos x \cdot 1 = \cos x$.

πŸ“ Amplitude and Period

For functions of the form $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$:

Example 3Amplitude and Period

Find the amplitude and period of $y = 3\sin(2x)$.

Solution: Amplitude $|3| = 3$. Period $= \frac{2\pi}{2} = \pi$.

πŸ–ΌοΈ Visual Comparison

Interactive graph below. Click buttons to see different trigonometric functions:

πŸ” Key Takeaway
Trigonometry is essential for calculus. Know the unit circle, exact values, and basic identities. They appear in limits, derivatives, and integrals constantly.

πŸ“ Practice Problems

Problem 1Evaluate

Find $\cos\left(\frac{2\pi}{3}\right)$.

Show solution

$\frac{2\pi}{3}$ is in Quadrant II. Reference angle: $\frac{\pi}{3}$. Cosine is negative in Quadrant II, so $\cos = -\frac{1}{2}$.

Problem 2Period

Find the period of $y = \cos\left(\frac{x}{2}\right)$.

Show solution

Period $= \frac{2\pi}{|B|} = \frac{2\pi}{1/2} = 4\pi$.

Problem 3Simplify

Simplify $\frac{\sin x}{\cos x} \cdot \cos x$.

Show solution

$\frac{\sin x}{\cos x} \cdot \cos x = \sin x$.

Problem 4Domain

Find the domain of $f(x) = \tan x$.

Show solution

All real numbers except $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

Problem 5Identity

Simplify $\cos^2 x - \sin^2 x$.

Show solution

$\cos^2 x - \sin^2 x = \cos(2x)$.

← Back to Module Page