How to Evaluate Each Trigonometric Function

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Trigonometric Functions

Trigonometric functions -also referred to as circular functions- are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the basic trigonometric functions.

How to Evaluate Trigonometric Functions Step by Step

Determine the reference angle. (It is the smallest angle that you can make from the terminal side of an angle with the \(x\)-axis.)
Find the reference angle's trigonometric function.

Example

Evaluate \(tan(\frac{5π}{4})\)

Solution

Rewrite the angles for \(\frac{5π}{4}\)

\(tan(\frac{5π}{4}) \ = \ tan(π \ + \ \frac{π}{4})\)

Utilize the \(tan\) periodicity: \(tan(θ \ + \ nπ) \ = \ tan(θ)\)

\(tan(π \ + \ \frac{π}{4}) \ = \ tan(\frac{π}{4}) \ = \ 1\)

Additional Angle Identities

\(Sin(π \ - \ θ) \ = \ sin(θ)\)
\(Cos(π \ - \ θ) \ = \ –cos(θ)\)
\(tan(π \ - \ θ) \ = \ – tan(θ)\)
\(Cosec(π \ - \ θ) \ = \ cosec(θ)\)
\(Sec(π \ - \ θ) \ = \ -sec(θ)\)
\(Cot(π \ - \ θ) \ = \ -cot(θ)\)

Evaluating Each Trigonometric Function

Think of this lesson as more than a rule to memorize. Evaluating Each Trigonometric Function 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 Evaluating Each Trigonometric Function

1) Evaluate \(sin \ \frac{π}{6}\).

2) Evaluate \(cos \ \frac{π}{3}\).

3) Evaluate \(tan \ \frac{π}{4}\).

4) Evaluate \(cosec \ \frac{π}{2}\).

5) Evaluate \(sec \ π\).

6) Evaluate \(cot \ \frac{3π}{4}\).

7) Evaluate \(sin \ \frac{4π}{3}\).

8) Evaluate \(cos \ \frac{5π}{4}\).

9) Evaluate \(tan \ \frac{2π}{3}\).

10) Evaluate \(sec \ \frac{11π}{6}\).

11) Evaluate \(cosec \ \frac{7π}{6}\).

12) Evaluate \(cot \ \frac{5π}{3}\).

13) Evaluate \(sin(-\frac{π}{3})\).

14) Evaluate \(cos(-\frac{5π}{6})\).

15) Evaluate \(tan \ \frac{17π}{4}\).

16) Evaluate \(cosec \ \frac{19π}{6}\).

17) Evaluate \(sec(-\frac{7π}{3})\).

18) Evaluate \(cot \ \frac{23π}{6}\).

19) Evaluate \(tan(-\frac{13π}{3})\).

20) Evaluate \(sin \ \frac{31π}{6}\).

Answers

1)Evaluate \(sin \ \frac{π}{6}\).

Use the special angle \(\frac{π}{6}\).

\(sin \ \frac{π}{6}=\frac{1}{2}\).

\(\color{red}{\sin \ \frac{π}{6} = \frac{1}{2}}\)

2)Evaluate \(cos \ \frac{π}{3}\).

Use the special angle \(\frac{π}{3}\).

\(cos \ \frac{π}{3}=\frac{1}{2}\).

\(\color{red}{\cos \ \frac{π}{3} = \frac{1}{2}}\)

3)Evaluate \(tan \ \frac{π}{4}\).

Use the special angle \(\frac{π}{4}\).

\(tan \ \frac{π}{4}=1\).

\(\color{red}{\tan \ \frac{π}{4} = 1}\)

4)Evaluate \(cosec \ \frac{π}{2}\).

\(sin \ \frac{π}{2}=1\).

Cosecant is the reciprocal of sine.

\(\color{red}{\csc \ \frac{π}{2} = 1}\)

5)Evaluate \(sec \ π\).

\(cos \ π=-1\).

Secant is the reciprocal of cosine.

\(\color{red}{\sec \ π = -1}\)

6)Evaluate \(cot \ \frac{3π}{4}\).

\(\frac{3π}{4}\) is in Quadrant II with reference angle \(\frac{π}{4}\).

Tangent is \(-1\), so cotangent is its reciprocal.

\(\color{red}{\cot \ \frac{3π}{4} = -1}\)

7)Evaluate \(sin \ \frac{4π}{3}\).

\(\frac{4π}{3}\) is in Quadrant III with reference angle \(\frac{π}{3}\).

Sine is negative in Quadrant III.

\(\color{red}{\sin \ \frac{4π}{3} = -\frac{\sqrt{3}}{2}}\)

8)Evaluate \(cos \ \frac{5π}{4}\).

\(\frac{5π}{4}\) is in Quadrant III with reference angle \(\frac{π}{4}\).

Cosine is negative in Quadrant III.

\(\color{red}{\cos \ \frac{5π}{4} = -\frac{\sqrt{2}}{2}}\)

9)Evaluate \(tan \ \frac{2π}{3}\).

\(\frac{2π}{3}\) is in Quadrant II with reference angle \(\frac{π}{3}\).

Tangent is negative in Quadrant II.

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

10)Evaluate \(sec \ \frac{11π}{6}\).

\(cos \ \frac{11π}{6}=\frac{\sqrt{3}}{2}\).

Take the reciprocal and rationalize.

\(\color{red}{\sec \ \frac{11π}{6} = \frac{2\sqrt{3}}{3}}\)

11)Evaluate \(cosec \ \frac{7π}{6}\).

\(sin \ \frac{7π}{6}=-\frac{1}{2}\).

Cosecant is the reciprocal of sine.

\(\color{red}{\csc \ \frac{7π}{6} = -2}\)

12)Evaluate \(cot \ \frac{5π}{3}\).

\(sin \ \frac{5π}{3}=-\frac{\sqrt{3}}{2}\) and \(cos \ \frac{5π}{3}=\frac{1}{2}\).

\(cot \theta=\frac{cos \theta}{sin \theta}\).

\(\color{red}{\cot \ \frac{5π}{3} = -\frac{\sqrt{3}}{3}}\)

13)Evaluate \(sin(-\frac{π}{3})\).

Sine is an odd function, so \(sin(-\theta)=-sin \theta\).

\(sin \ \frac{π}{3}=\frac{\sqrt{3}}{2}\).

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

14)Evaluate \(cos(-\frac{5π}{6})\).

Cosine is an even function, so \(cos(-\theta)=cos \theta\).

\(cos \ \frac{5π}{6}=-\frac{\sqrt{3}}{2}\).

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

15)Evaluate \(tan \ \frac{17π}{4}\).

Subtract \(4π=\frac{16π}{4}\) to get \(\frac{π}{4}\).

\(tan \ \frac{π}{4}=1\).

\(\color{red}{\tan \ \frac{17π}{4} = 1}\)

16)Evaluate \(cosec \ \frac{19π}{6}\).

Subtract \(2π\) to get \(\frac{7π}{6}\).

\(sin \ \frac{7π}{6}=-\frac{1}{2}\), so its reciprocal is \(-2\).

\(\color{red}{\csc \ \frac{19π}{6} = -2}\)

17)Evaluate \(sec(-\frac{7π}{3})\).

Add \(2π\) to get \(-\frac{π}{3}\).

\(cos(-\frac{π}{3})=\frac{1}{2}\), so the reciprocal is \(2\).

\(\color{red}{\sec(-\frac{7π}{3}) = 2}\)

18)Evaluate \(cot \ \frac{23π}{6}\).

Subtract \(2π=\frac{12π}{6}\) to get \(\frac{11π}{6}\).

\(cot \ \frac{11π}{6}=\frac{\sqrt{3}/2}{-1/2}\).

\(\color{red}{\cot \ \frac{23π}{6} = -\sqrt{3}}\)

19)Evaluate \(tan(-\frac{13π}{3})\).

Add \(4π=\frac{12π}{3}\) to get \(-\frac{π}{3}\).

\(tan(-\frac{π}{3})=-\sqrt{3}\).

\(\color{red}{\tan(-\frac{13π}{3}) = -\sqrt{3}}\)

20)Evaluate \(sin \ \frac{31π}{6}\).

Subtract \(4π=\frac{24π}{6}\) to get \(\frac{7π}{6}\).

\(sin \ \frac{7π}{6}=-\frac{1}{2}\).

\(\color{red}{\sin \ \frac{31π}{6} = -\frac{1}{2}}\)

How to Evaluate Each Trigonometric Function