1. Introduction
The six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) can be used to find the length of the sides of a triangle or the measure of an angle if the length of two sides is given. Previously these functions could only be applied to angles up to 90°. However, by using radians and the unit circle, these functions can be applied to any angle.
5.1.4: Evaluating Trigonometric Functions

2. Key Concepts
Recall that sine is the ratio of the length of the opposite side to the length of the hypotenuse, cosine is the ratio of the length of the adjacent side to the length of the hypotenuse, and tangent is the ratio of the length of the opposite side to the length of the adjacent side. (You may have used the mnemonic device SOHCAHTOA to help remember these relationships: Sine equals the Opposite side over the Hypotenuse, Cosine equals the Adjacent side over the Hypotenuse, and Tangent equals the Opposite side over the Adjacent side.)
5.1.4: Evaluating Trigonometric Functions

3. Key Concepts, continued
Three other trigonometric functions, cosecant, secant, and cotangent, are reciprocal functions of the first three. Cosecant is the reciprocal of the sine function, secant is the reciprocal of the cosine function, and cotangent is the reciprocal of the tangent function.
5.1.4: Evaluating Trigonometric Functions

4. Key Concepts, continued
The cosecant of θ = csc θ = ;
The secant of θ = sec θ =
The cotangent of θ = cot θ =
5.1.4: Evaluating Trigonometric Functions

5. Key Concepts, continued
The quadrant in which the terminal side is located determines the sign of the trigonometric functions. In Quadrant I, all the trigonometric functions are positive. In Quadrant II, the sine and its reciprocal, the cosecant, are positive and all the other functions are negative. In Quadrant III, the tangent and its reciprocal, the cotangent, are positive, and all other functions are negative. In Quadrant IV, the cosine and its reciprocal, the secant, are positive, and all other functions are negative.
5.1.4: Evaluating Trigonometric Functions

6. Key Concepts, continued
You can use a mnemonic device to remember in which quadrants the functions are positive: All Students Take Calculus (ASTC).
5.1.4: Evaluating Trigonometric Functions

7. Key Concepts, continued
However, instead of memorizing this, you can also think it through each time, considering whether the opposite and adjacent sides of the reference angle are positive or negative in each quadrant.
To find a trigonometric function of an angle given a point on its terminal side, first visualize a triangle using the reference angle. The x-coordinate becomes the length of the adjacent side and the y-coordinate becomes the length of the opposite side. The length of the hypotenuse can be found using the Pythagorean Theorem. Determine the sign by remembering the ASTC pattern or by considering the signs of the x- and y-coordinates.
5.1.4: Evaluating Trigonometric Functions

8. Key Concepts, continued
To find the trigonometric functions of special angles, first find the reference angle and then use the pattern to determine the ratio.
For angles larger than 2π radians (360°), subtract 2π radians (360°) to find a coterminal angle, an angle that shares the same terminal side, that is less than 2π radians (360°). Repeat if necessary.
For negative angles, find the reference angle and then apply the same method.
5.1.4: Evaluating Trigonometric Functions

9. Common Errors/Misconceptions
using the incorrect trigonometric ratio
forgetting to consider whether the trigonometric ratios are negative
mistaking the quadrants in which each trigonometric function is positive
5.1.4: Evaluating Trigonometric Functions

10. Guided Practice Example 2
Find sin θ if θ is a positive angle in standard position with a terminal side that passes through the point (5, –2). Give an exact answer.
5.1.4: Evaluating Trigonometric Functions

11. Guided Practice: Example 2, continued
Sketch the angle and draw in the triangle associated with the reference angle.
Recall that a positive angle is created by rotating counterclockwise around the origin of the coordinate plane.
Plot (5, –2) on a coordinate plane and draw the terminal side extending from the origin through that point.
5.1.4: Evaluating Trigonometric Functions

12. Guided Practice: Example 2, continued
The reference angle is the angle the terminal side makes with the x-axis.
5.1.4: Evaluating Trigonometric Functions

13. Guided Practice: Example 2, continued
Notice that θ is nearly 360°, so the reference angle is in the fourth quadrant.
The magnitude of the x-coordinate is the length of the adjacent side and the magnitude of the y-coordinate is the length of the opposite side. The hypotenuse can be found using the Pythagorean Theorem. Determine the sign of sin θ by recalling the ASTC pattern or by considering the signs of the x- and y-coordinates.
5.1.4: Evaluating Trigonometric Functions

14. Guided Practice: Example 2, continued
Find the length of the opposite side and the length of the hypotenuse.
Sine is the ratio of the length of the opposite side to the length of the hypotenuse; therefore, these two lengths must be determined.
The length of the opposite side is the magnitude of the y-coordinate, 2.
5.1.4: Evaluating Trigonometric Functions

15. Guided Practice: Example 2, continued
Since the opposite side length is known to be 2 and the adjacent side length, 5, can be determined from the sketch, the hypotenuse can be found by using the Pythagorean Theorem.
The length of the hypotenuse is units.
c2 = a2 + b2
Pythagorean Theorem
c2 = (2)2 + (5)2
Substitute 2 for a and 5 for b.
c2 =
Simplify the exponents.
c2 = 29
Add.
Take the square root of both sides.
5.1.4: Evaluating Trigonometric Functions

16. Guided Practice: Example 2, continued Find sin θ.
Now that the lengths of the opposite side and the hypotenuse are known, substitute these values into the sine ratio to determine sin θ.
Sine ratio
Substitute 2 for the opposite side and for the hypotenuse.
5.1.4: Evaluating Trigonometric Functions

17. ✔ Guided Practice: Example 2, continued Rationalize the denominator.
According to ASTC, in Quadrant IV only the cosine and secant are positive. The sine is negative.
For a positive angle θ in standard position with a terminal side that passes through the point (5, –2),
Rationalize the denominator. ✔
5.1.4: Evaluating Trigonometric Functions

18. Guided Practice: Example 2, continued
5.1.4: Evaluating Trigonometric Functions

19. Guided Practice Example 4 Given , if θ is in Quadrant I, find cot θ.
5.1.4: Evaluating Trigonometric Functions

20. Guided Practice: Example 4, continued
Sketch an angle in Quadrant I, draw the associated triangle, and label the sides with the given information.
Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. Since , 4 is the length of the adjacent side and 5 is the length of the hypotenuse.
5.1.4: Evaluating Trigonometric Functions

21. Guided Practice: Example 4, continued
5.1.4: Evaluating Trigonometric Functions

22. Guided Practice: Example 4, continued
Use the Pythagorean Theorem to find the length of the opposite side.
Since the lengths of two sides of the triangle are given, substitute these values into the Pythagorean Theorem and solve for the missing side length.
5.1.4: Evaluating Trigonometric Functions

23. Guided Practice: Example 4, continued
The length of the opposite side is 3 units.
c2 = a2 + b2
Pythagorean Theorem
(5)2 = (4)2 + b2
Substitute 5 for c and 4 for a.
25 = 16 + b2
Simplify the exponents.
9 = b2
Subtract 16 from both sides.
3 = b
Take the square root of both sides.
5.1.4: Evaluating Trigonometric Functions

24. Guided Practice: Example 4, continued Find the cotangent.
Use the values from the triangle to determine the cotangent.
Cotangent ratio
Substitute 4 for the adjacent side and 3 for the opposite side.
5.1.4: Evaluating Trigonometric Functions

25. ✔ Guided Practice: Example 4, continued
In Quadrant I, all trigonometric ratios are positive, which coincides with the answer found.
Given , for an angle θ in Quadrant I, ✔
5.1.4: Evaluating Trigonometric Functions

26. Guided Practice: Example 4, continued
5.1.4: Evaluating Trigonometric Functions