How to Find Trigonometric Ratios of General Angles

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:

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

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) $$cosec \ \frac{8π}{3} \ =$$

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) $$cosec \ \frac{8π}{3} \ =$$

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

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