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 |
\(0^\circ\) |
\(30^\circ \) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
| \(sinâ¡θ\) |
\(0\) |
\(\frac{1}{2}\) |
\(\frac{\sqrt2}{2}\) |
\(\frac{\sqrt3}{2}\) |
\(1\) |
| \(cosâ¡θ\) |
\(1\) |
\(\frac{\sqrt3}{2}\) |
\(\frac{\sqrt2}{2}\) |
\(\frac{1}{2}\) |
\(0\) |
| \(tanâ¡θ\) |
\(0\) |
\(\frac{\sqrt3}{3}\) |
\(1\) |
\(\sqrt{3}\) |
\(∞\) |
| \(cotθ\) |
\(∞\) |
\(\sqrt{3}\) |
\(1\) |
\(\frac{\sqrt3}{3}\) |
\(0\) |
| \(secθ\) |
\(1\) |
\(\frac{2}{\sqrt{3}}\) |
\(\sqrt{2}\) |
\(2\) |
\(∞\) |
| \(cosecθ\) |
\(∞\) |
\(2\) |
\(\sqrt{2}\) |
\(\frac{2}{\sqrt{3}}\) |
\(1\) |
Trig ratios of General Angles
Think of this lesson as more than a rule to memorize. Trig ratios of General Angles is about angles, triangles, radians, and circular motion. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
Trig ratios connect angles to side lengths. In a right triangle, \(\sin\theta=\frac{opposite}{hypotenuse}\), \(\cos\theta=\frac{adjacent}{hypotenuse}\), and \(\tan\theta=\frac{opposite}{adjacent}\).
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Identify the input value or expression.
- Substitute carefully using parentheses.
- Simplify one operation at a time.
- Check domain restrictions such as zero denominators or even roots.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Trig ratios of General Angles
1) Evaluate \(sin \ 30°\).
2) Evaluate \(cos \ 120°\).
3) Evaluate \(tan \ 225°\).
4) Evaluate \(cot \ 330°\).
5) Evaluate \(sec \ 240°\).
6) Evaluate \(cosec \ \frac{5π}{6}\).
7) Evaluate \(sin \ \frac{7π}{4}\).
8) Evaluate \(cos(-60°)\).
9) Evaluate \(tan \ \frac{11π}{6}\).
10) Evaluate \(sec \ \frac{3π}{4}\).
11) Evaluate \(cot(-\frac{π}{4})\).
12) Evaluate \(cosec \ \frac{4π}{3}\).
13) Evaluate \(sin \ 765°\).
14) Evaluate \(cos(-210°)\).
15) Evaluate \(tan \ \frac{5π}{3}\).
16) Evaluate \(sec \ \frac{17π}{6}\).
17) Evaluate \(cot \ \frac{13π}{4}\).
18) Evaluate \(cosec(-\frac{7π}{6})\).
19) Evaluate \(tan(-585°)\).
20) Evaluate \(sec \ \frac{19π}{3}\).
1)Evaluate \(sin \ 30°\).
Use the unit-circle value for \(30°\).
\(sin \ 30°=\frac{1}{2}\).
\(\color{red}{\frac{1}{2}}\)
2)Evaluate \(cos \ 120°\).
Write \(120°=180°-60°\), so the angle is in Quadrant II.
Cosine is negative in Quadrant II, and the reference value is \(cos \ 60°=\frac{1}{2}\).
\(\color{red}{\cos \ 120° = -\frac{1}{2}}\)
3)Evaluate \(tan \ 225°\).
Write \(225°=180°+45°\), so the angle is in Quadrant III.
Tangent is positive in Quadrant III, and \(tan \ 45°=1\).
\(\color{red}{\tan \ 225° = 1}\)
4)Evaluate \(cot \ 330°\).
Write \(330°=360°-30°\), so the angle is in Quadrant IV.
\(sin \ 330°=-\frac{1}{2}\) and \(cos \ 330°=\frac{\sqrt{3}}{2}\).
\(cot \theta=\frac{cos \theta}{sin \theta}\), so \(\color{red}{\cot \ 330° = -\sqrt{3}}\).
5)Evaluate \(sec \ 240°\).
Write \(240°=180°+60°\), so the angle is in Quadrant III.
\(cos \ 240°=-\frac{1}{2}\).
\(sec \theta=\frac{1}{cos \theta}\), so \(\color{red}{\sec \ 240° = -2}\).
6)Evaluate \(cosec \ \frac{5π}{6}\).
The angle \(\frac{5π}{6}\) is in Quadrant II with reference angle \(\frac{π}{6}\).
\(sin \ \frac{5π}{6}=\frac{1}{2}\).
\(cosec \theta=\frac{1}{sin \theta}\), so \(\color{red}{\csc \ \frac{5π}{6} = 2}\).
7)Evaluate \(sin \ \frac{7π}{4}\).
The angle \(\frac{7π}{4}\) is in Quadrant IV with reference angle \(\frac{π}{4}\).
Sine is negative in Quadrant IV, and \(sin \ \frac{π}{4}=\frac{\sqrt{2}}{2}\).
\(\color{red}{\sin \ \frac{7π}{4} = -\frac{\sqrt{2}}{2}}\)
8)Evaluate \(cos(-60°)\).
Cosine is an even function, so \(cos(-\theta)=cos \theta\).
Therefore \(cos(-60°)=cos \ 60°\).
\(\color{red}{\cos(-60°) = \frac{1}{2}}\)
9)Evaluate \(tan \ \frac{11π}{6}\).
The angle \(\frac{11π}{6}\) is in Quadrant IV with reference angle \(\frac{π}{6}\).
Tangent is negative in Quadrant IV, and \(tan \ \frac{π}{6}=\frac{\sqrt{3}}{3}\).
\(\color{red}{\tan \ \frac{11π}{6} = -\frac{\sqrt{3}}{3}}\)
10)Evaluate \(sec \ \frac{3π}{4}\).
The angle \(\frac{3π}{4}\) is in Quadrant II with reference angle \(\frac{π}{4}\).
\(cos \ \frac{3π}{4}=-\frac{\sqrt{2}}{2}\).
Taking the reciprocal gives \(\color{red}{\sec \ \frac{3π}{4} = -\sqrt{2}}\).
11)Evaluate \(cot(-\frac{π}{4})\).
Cotangent has period \(π\), and \(-\frac{π}{4}\) has reference angle \(\frac{π}{4}\).
\(tan(-\frac{π}{4})=-1\).
Since \(cot \theta=\frac{1}{tan \theta}\), \(\color{red}{\cot(-\frac{π}{4}) = -1}\).
12)Evaluate \(cosec \ \frac{4π}{3}\).
The angle \(\frac{4π}{3}\) is in Quadrant III with reference angle \(\frac{π}{3}\).
\(sin \ \frac{4π}{3}=-\frac{\sqrt{3}}{2}\).
Taking the reciprocal and rationalizing gives \(\color{red}{\csc \ \frac{4π}{3} = -\frac{2\sqrt{3}}{3}}\).
13)Evaluate \(sin \ 765°\).
Subtract \(720°\) to find a coterminal angle: \(765°-720°=45°\).
So \(sin \ 765°=sin \ 45°\).
\(\color{red}{\sin \ 765° = \frac{\sqrt{2}}{2}}\)
14)Evaluate \(cos(-210°)\).
Add \(360°\): \(-210°+360°=150°\).
\(150°\) is in Quadrant II with reference angle \(30°\), where cosine is negative.
\(\color{red}{\cos(-210°) = -\frac{\sqrt{3}}{2}}\)
15)Evaluate \(tan \ \frac{5π}{3}\).
The angle \(\frac{5π}{3}\) is in Quadrant IV with reference angle \(\frac{π}{3}\).
Tangent is negative in Quadrant IV, and \(tan \ \frac{π}{3}=\sqrt{3}\).
\(\color{red}{\tan \ \frac{5π}{3} = -\sqrt{3}}\)
16)Evaluate \(sec \ \frac{17π}{6}\).
Subtract \(2π=\frac{12π}{6}\): \(\frac{17π}{6}-\frac{12π}{6}=\frac{5π}{6}\).
\(cos \ \frac{5π}{6}=-\frac{\sqrt{3}}{2}\).
The reciprocal is \(\color{red}{\sec \ \frac{17π}{6} = -\frac{2\sqrt{3}}{3}}\).
17)Evaluate \(cot \ \frac{13π}{4}\).
Subtract \(2π=\frac{8π}{4}\): \(\frac{13π}{4}-\frac{8π}{4}=\frac{5π}{4}\).
At \(\frac{5π}{4}\), \(sin\) and \(cos\) are both \(-\frac{\sqrt{2}}{2}\).
\(cot \theta=\frac{cos \theta}{sin \theta}\), so \(\color{red}{\cot \ \frac{13π}{4} = 1}\).
18)Evaluate \(cosec(-\frac{7π}{6})\).
Add \(2π=\frac{12π}{6}\): \(-\frac{7π}{6}+\frac{12π}{6}=\frac{5π}{6}\).
\(sin \ \frac{5π}{6}=\frac{1}{2}\).
Taking the reciprocal gives \(\color{red}{\csc(-\frac{7π}{6}) = 2}\).
19)Evaluate \(tan(-585°)\).
Add \(540°\), a multiple of the \(180°\) tangent period: \(-585°+540°=-45°\).
\(tan(-45°)=-tan \ 45°\).
\(\color{red}{\tan(-585°) = -1}\)
20)Evaluate \(sec \ \frac{19π}{3}\).
Subtract \(6π=\frac{18π}{3}\): \(\frac{19π}{3}-\frac{18π}{3}=\frac{π}{3}\).
\(cos \ \frac{π}{3}=\frac{1}{2}\).
Taking the reciprocal gives \(\color{red}{\sec \ \frac{19π}{3} = 2}\).
Trig ratios of General Angles Practice Quiz
