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:
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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 |
\(0^\circ\) |
\(30^\circ \) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
| \(sinθ\) |
\(0\) |
\(\frac{1}{2}\) |
\(\frac{\sqrt2}{2}\) |
\(\frac{\sqrt3}{2}\) |
\(1\) |
| \(cosθ\) |
\(1\) |
\(\frac{\sqrt3}{2}\) |
\(\frac{\sqrt2}{2}\) |
\(\frac{1}{2}\) |
\(0\) |
| \(tanθ\) |
\(0\) |
\(\frac{\sqrt3}{3}\) |
\(1\) |
\(\sqrt{3}\) |
\(∞\) |
| \(cotθ\) |
\(∞\) |
\(\sqrt{3}\) |
\(1\) |
\(\frac{\sqrt3}{3}\) |
\(0\) |
| \(secθ\) |
\(1\) |
\(\frac{2}{\sqrt{3}}\) |
\(\sqrt{2}\) |
\(2\) |
\(∞\) |
| \(cosecθ\) |
\(∞\) |
\(2\) |
\(\sqrt{2}\) |
\(\frac{2}{\sqrt{3}}\) |
\(1\) |
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) \(cosec \ \frac{8π}{3} \ =\)
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) \(cosec \ \frac{8π}{3} \ =\)
\(\color{red}{cosec \ \frac{8π}{3} \ = \ cosec \ (3π \ - \ \frac{π}{3}) \ = \ cosec \ \frac{π}{3} \ = \frac{2\sqrt{3}}{3}}\)
Trig ratios of General Angles Practice Quiz
