Trigonometry

• Trigono = Triangle
• Metry = Measure

$\quad$

• $\text{Cosine} \to \cos \theta = \frac{\overline{AB}}{\overline{AC}}$

• $\text{Sine} \to \sin \theta = \frac{\overline{BC}}{\overline{AC}}$

• $\tan \theta = \frac{\overline{BC}}{\overline{AB}}$

• $\tan 30^{\circ}= \frac{{H}} {{S+10}} = \frac{1}{\sqrt3}$

• $\tan 60^{\circ}= \frac{{H}} {{S}} =\sqrt3$

• $AC = \text{Hypotenuse}$
• $BC = \text{Adjacent side}$
• $AB = \text{Opposite side}$

Anti-clockwise rotation $\Rightarrow$ positive angle

Clockwise rotation $\Rightarrow$ negative angle

There are two popular ways to measure angle:

1)In Degree

2)In Radian

• $1$ complete revolution $= 360^{\circ}$
• $1^{\circ} = (\frac{1}{360})^{\text{th}}$ of a complete revolution.

Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

1 complete Revolution = 2π RAD

• Radius of Inner Circle - 1 unit
• Radius of Outer Circle - R units.
  $\quad$

$\Rightarrow$ 2 π R = 2 π $x$

$\Rightarrow$ $x$ = R

$\quad$

• l = θ R

• 1 Complete revolution = 360° = 2 $\pi$ RAD
  $\therefore$ 1 RAD = $\frac{{360°}} {{2 \pi}}$ = 57.27° ($\pi$ = $\frac{22}{7}$)
• 360° = 2$\pi$ RAD
  $\Rightarrow \ 1^{\circ} = \frac{{2 \pi}} {{360}}$ RAD

1. Find the value of $\frac{{\pi}} {{4}}$ RAD .

$\frac{{\pi}} {{4}}$ RAD = $\frac{{\pi}} {{4}} \times \frac{{360}^{\circ}} {{2\pi}}= 45°$

2. Find the value of $135°.$

$135° = 135° \times \frac{{2\pi}} {{360}}$ RAD

$= \biggl(\frac{{3\pi }} {{4}}\biggl)$RAD

The minute hand of a watch is 5 cm long. How far does its tip move in 42 minutes?

Solution:

1 complete revolution $=$ 2π RAD

$\frac{42}{60}$ of 1 revolution

$= \frac {42}{60} {2 \pi}$ RAD = ${1.4}{\pi}$ RAD

We know that, l $=$ θ R

l $= {1.4}{\pi} \ 5 \text{cm} \\ \ =22 \ \text{cm}$

• $\sin (x) = b$
• $\cos (x) = a$
• $x \in \R, \$ Domain $= \R$
• Range of Sine: $-1 ≤ a ≤ 1$
• Range of Cosine $-1 ≤ b ≤ 1$

$\sin \ x = {b} \$ and $\ \cos \ x = {a}$

$OA^2 + AP^2 = OP^2$

$a^2+b^2=1$

$\cos^2x + \sin^2x = 1$

$x=0, x= π$

$\sin(x) = \sin(x+k.2π)$

for any integer k.

$\sin x = 0 \Leftrightarrow x = kπ$

Thank you