How to Find Trigonometric Ratios of General Angles

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Trig ratios of General Angles

Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant are trigonometric ratios. For these trigonometric ratios, the standard angles are $0, \ 30, \ 45, \ 60,$ and $90$ degrees. These angles can also be shown using radians, such as $0, \ \frac{π}{6}, \ \frac{π}{4}, \ \frac{π}{3},$ and $\frac{π}{2}$ . In trigonometry, these angles are most regularly and frequently used. To solve many problems, you need to know the values of these trigonometry angles.

Here are the trigonometry ratios for a particular $θ$ angle:

$Sin \ θ$	$\frac{Opposite \ Side \ to \ θ}{Hypotenuse}$
$Cos \ θ$	$\frac{Adjacent \ Side \ to \ θ}{Hypotenuse}$
$Tan \ θ$	$\frac{Opposite \ Side \ to \ θ}{Adjacent \ Side \ to \ θ}$ OR $\frac{Sin \ θ}{Cos \ θ}$
$Cot \ θ$	$\frac{Adjacent \ Side \ to \ θ}{Opposite \ Side \ to \ θ}$ OR $\frac{Cos \ θ}{Sin \ θ} \ = \ \frac{1}{tan \ θ}$
$Sec \ θ$	$\frac{Hypotenuse}{Adjacent \ Side \ to \ θ}$ OR $\frac{1}{cos \ θ}$
$Cosec \ θ$	$\frac{Hypotenuse}{Opposite \ Side \ to \ θ}$ OR $\frac{1}{sin \ θ}$

Finding Trigonometric Ratios

Consider this right triangle:

Trig ratios of General Angles

The trigonometric ratios of $∠ \ C$ are:

Sine: The ratio of an angle's perpendicular (opposite) side to its hypotenuse is known as its sine.
Cosine: The ratio of the side adjacent to the angle to the hypotenuse is known as the cosine of an angle.
Tangent: The tangent of an angle is the ratio between the side opposite the angle and the side adjacent to it.
cotangent: tangent's multiplicative inverse is cotangent.
Cosecant: Sine's multiplicative inverse is cosecant.
Secant: cosine's multiplicative inverse is Secant.

In the order they are listed, the above ratios are written as $sin, \ cos, \ tan, \ cosec, \ sec,$ and $tan$ . So, in the case of $Δ \ ABC$ , the ratios are:

$Sin \ C \ = \ \frac{Opposite \ Side \ to \ ∠ \ C}{Hypotenuse} \ = \ \frac{AB}{AC}$

$Cos \ C \ = \ \frac{Adjacent \ Side \ to \ ∠ \ C}{Hypotenuse} \ = \ \frac{BC}{AC}$

$tan \ C \ = \ \frac{Opposite \ Side \ to \ ∠ \ C}{Adjacent \ Side \ to \ ∠ \ C} \ = \ \frac{AB}{BC} \ = \ \frac{Sin \ C}{Cos \ C}$

$Cot \ C \ = \ \frac{Adjacent \ Side \ to \ ∠ \ C}{Opposite \ Side \ to \ ∠ \ C} \ = \ \frac{BC}{AB} \ = \ \frac{1}{tan \ C}$

$Cosec \ C \ = \ \frac{Hypotenuse}{Opposite \ Side \ to \ ∠ \ C} \ = \ \frac{AC}{AB} \ = \ \frac{1}{Sin \ C}$

$Sec \ C \ = \ \frac{Hypotenuse}{Adjacent \ Side \ to \ ∠ \ C} \ = \ \frac{AC}{BC} \ = \ \frac{1}{Cos \ C}$

Table of Trigonometric Ratios

Below are the trigonometric ratios for certain angles, such as 0, 30, 45, 60, and 90 degrees.

θ	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$sin⁡θ$	$0$	$\frac{1}{2}$	$\frac{\sqrt2}{2}$	$\frac{\sqrt3}{2}$	$1$
$cos⁡θ$	$1$	$\frac{\sqrt3}{2}$	$\frac{\sqrt2}{2}$	$\frac{1}{2}$	$0$
$tan⁡θ$	$0$	$\frac{\sqrt3}{3}$	$1$	$\sqrt{3}$	$∞$
$cotθ$	$∞$	$\sqrt{3}$	$1$	$\frac{\sqrt3}{3}$	$0$
$secθ$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	$∞$
$cosecθ$	$∞$	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$

Free printable Worksheets

Exercises for Trig ratios of General Angles

1) $sin \ 120° \ =$

2) $cos \ \frac{7π}{6} \ =$

3) $tan \ 585° \ =$

4) $cot \ 450° \ =$

5) $sec \ 45° \ =$

6) $cos \ \frac{5π}{6} \ =$

7) $cosec \ \frac{2π}{3} \ =$

8) $sin \ \frac{11π}{4} \ =$

9) $cos \ \frac{11π}{4} \ =$

10) $cosec \ \frac{8π}{3} \ =$

Answers

1) $sin \ 120° \ =$

$\color{red}{sin \ 120° \ = \ sin \ (180° \ - \ 60°) \ = \ sin \ 60° \ = \frac{\sqrt{3}}{2}}$

2) $cos \ \frac{7π}{6} \ =$

$\color{red}{cos \ \frac{7π}{6} \ = \ cos \ (π \ + \ \frac{π}{6}) \ = \ -cos \ \frac{π}{6} \ = -\frac{\sqrt{3}}{2}}$

3) $tan \ 585° \ =$

$\color{red}{tan \ 585° \ = \ tan \ (3(180°) \ + \ 45°) \ = \ tan \ 45° \ = \ 1}$

4) $cot \ 450° \ =$

$\color{red}{cot \ 450° \ = \ cot \ (360° \ + \ 90°) \ = \ cot \ 90° \ = \ 0}$

5) $sec \ 45° \ =$

$\color{red}{sec \ 45° \ = \ \sqrt{2}}$

6) $cos \ \frac{5π}{6} \ =$

$\color{red}{cos \ \frac{5π}{6} \ = \ cos \ (π \ - \ \frac{π}{6}) \ = \ -cos \ \frac{π}{6} \ = -\frac{\sqrt{3}}{2}}$

7) $cosec \ \frac{2π}{3} \ =$

$\color{red}{cosec \ \frac{2π}{3} \ = \ cosec \ (π \ - \ \frac{π}{3}) \ = \ cosec \ \frac{π}{3} \ = \frac{2}{\sqrt{3}}}$

8) $sin \ \frac{11π}{4} \ =$

$\color{red}{sin \ \frac{11π}{4} \ = \ sin \ (3π \ - \ \frac{π}{4}) \ = \ sin \ \frac{π}{4} \ = \frac{\sqrt2}{2}}$

9) $cos \ \frac{11π}{4} \ =$

$\color{red}{cos \ \frac{11π}{4} \ = \ cos \ (3π \ - \ \frac{π}{4}) \ = \ -cos \ \frac{π}{4} \ = -\frac{\sqrt2}{2}}$

10) $cosec \ \frac{8π}{3} \ =$

$\color{red}{cosec \ \frac{8π}{3} \ = \ cosec \ (3π \ - \ \frac{π}{3}) \ = \ cosec \ \frac{π}{3} \ = \frac{2\sqrt{3}}{3}}$

How to Find Trigonometric Ratios of General Angles