Find the values of the six trigonometric functions for the angle θ worksheet

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Summary: In this section, you will:

• Evaluate trigonometric functions of any angle.
• Find reference angles.

The London Eye is a Ferris wheel with a diameter of 394 feet. By combining the ideas of the unit circle and right triangles, the location of any capsule on the Eye can be described with trigonometry.

Trigonometric Functions of Angles on Circles with r ≠ 1

Lesson 4-02 looked at the unit circle. Lesson 4-03 explored right triangle trigonometry. To combine these ideas, consider a circle where r ≠ 1. Pick a point on the circle. A right triangle can be drawn to the point where one acute angle is at the point, the other acute angle is at the origin, and the right angle is on the x-axis.

Comparing the unit circle formulas and the right triangle formulas develops the formulas for any angle. For example, consider sin θ.

sinθ=yUnit Circlesinθ=opphypRight Trianglesinθ=yrApply the right triangle formula for the acute angle by the origin.

Notice the last equation matches the unit circle formula with r = 1. All the unit circle formulas can be similarly modified.

Trigonometric Functions of Any Angle

where θ is an angle in standard position with point (x, y) on the terminal side and r=x2+y2

Example 1: Evaluate Trigonometric Functions

Let (−4, 3) be a point on the terminal side of angle θ. Evaluate the six trigonometric functions of θ.

Find r.

r=x2+y2r=-42 +32r=5

Now use the trigonometric formulas.

sinθ=yr=35cscθ=ry= 53cosθ=xr=-45secθ=rx=-54tanθ=yx=-34cotθ= yr=-43

Try It 1

If (4, −8) is a point on the terminal side of angle α in standard position, evaluate the six trigonometric functions of α.

sinα=-25 5cscα=-52cosα=55secα=5tanα=-2cotα=-12

Example 2: Evaluate Trigonometric Functions of Quadrantal Angles

Evaluate cos 270° and csc π.

270° and π radians terminal sides are both on an axis. Start by choosing a point on the terminal sides of the angle.

Now apply the trigonometric formulas with r = 1.

cosθ=xrcscθ=rycos270°=01=0cscπ=10=undefined

Try It 2

Evaluate sin 90° and cot 0.

1; undefined

Signs of Trigonometric Functions in the Quadrants

By filling in the negative signs for x and y from the quadrants into the trigonometric formulas a pattern develops. For example, consider sine and cosine in quadrant II where the x is negative and y is positive.

All the trigonometric functions' signs can be similarly determined for all four quadrants. Figure 5 shows which trigonometric functions are positive in each quadrant.

Example 3: Evaluate Trigonometric Functions

If cosθ=-817 and sin θ < 0, find tan θ and csc θ.

cosθ=xr=-817

Since the r is always positive, r = 17 and x = −8. Use the Pythagorean Theorem to find y.

x2+y2=r2-82+y2=172y2=225 y=±15

Since sin θ < 0 and sinθ=yr, y must be negative. So, y = −15.

Now it is known that x = −8, y = −15, and r = 17. Since both x and y are negative, the angle terminates in quadrant III where tan θ and cot θ are positive. We could have also looked for a quadrant where both sin θ and cos θ were negative which is quadrant III. Now fill in the trigonometric formulas.

sinθ=yr=-1517cscθ=ry =-1715cosθ=xr=-817secθ=rx=-178tanθ=yx=158cotθ =yr=815

Try It 3

If sinθ=-53 and cos θ > 0, find tan θ and cos θ.

tanθ= -52; cosθ=23

Reference Angles

Since the formula from the unit circle and the right triangles give the same expressions for example 1, the acute angle by the origin in the triangle and the angle in standard position must give the same values of the trigonometric functions. Those acute angles are useful and called reference angles.

Reference Angles

The reference angle is the angle between the terminal side of an angle in standard position and the nearest x-axis. Reference angles are always less than π2. The values of the trigonometric functions of the angle in standard position equal values of the trigonometric functions of the reference angle with the appropriate negative signs for the quadrant.

1. Determine the measure of the angle whose terminal side is on the x-axis nearest the terminal side of the given angle.
2. Subtract the measures of the given angle and the x-axis angle.

Example 4: Find a Reference Angle

Find the reference angle for a) 7π6, b) 2π3, c) π4, and d) 7π4.

1. 
   Figure 7: \(\frac{7π}{6}\)

   It is easiest to start by stetching a graph of the angle in standard position like in figure 7.

   The x-axis angle nearest the terminal side of angle θ is π. Subtract π and 7π6 to find the reference angle.

   7π6-π7π6-6π 6=π6

   The reference angle is π6.

2. 
   Figure 8: \(\frac{2π}{3}\)

   The graph of 2π3 is in figure 8.

   The x-axis angle nearest the terminal side of angle θ is π. Subtract π and 2π3 to find the reference angle.

   π-2 π33π3-2π3=π3

   The reference angle is π3.

3. 
   Figure 9: \(\frac{π}{4}\)

   The graph of π4 is in figure 9.

   The x-axis angle nearest the terminal side of angle θ is 0. Subtract 0 and π4 to find the reference angle.

   π4-0=π4

   The reference angle is π4.

4. 
   Figure 10: \(\frac{7π}{4}\)

   The graph of 7π4 is in figure 10.

   The x-axis angle nearest the terminal side of angle θ is 2π. Subtract 2π and 7π4 to find the reference angle.

   2π-7π48π4-7π4=π4

   The reference angle is π4.

Try it 4

Find the reference angle for 5π3.

π3

Trigonometric Functions of Real Numbers

All the ideas from this lesson can be combined to evaluate trigonometric functions of any real number.

Evaluate Trigonometric Functions of Any Real Number

1. Consider the number to be an angle, θ.
2. If θ is not between 0 and 2π, find a coterminal angle between 0 and 2π by adding or subtracting 2π.
3. Find the reference angle.
4. Evaluate the trigonometric function of the reference angle using special right triangles (lesson 4-04) or the unit circle (lesson 4-02).
5. Apply a negative sign as needed based on the quadrant θ is in.

Example 5: Use Reference Angles

Evaluate a) cos 7π6, b) sin2π3, c) tan13π4, and d) sin-7π 4.

1. 7π6 is between 0 and 2π, so start by finding the reference angle. Example 4 found the reference angle of 7π6 is π6. Evaluate the function of the reference angle using special right triangles or the unit circle.

   cosπ6=32

   7π 6 is in quadrant III and cosine is negative in quadrant III, so cos7π6=-32.

2. 2π3 is between 0 and 2π, so start by finding the reference angle. Example 4 found the reference angle of 2π3 is π3. Evaluate the function of the reference angle using special right triangles or the unit circle.

   sinπ 3=32

   2π3 is in quadrant II and sine is positive in quadrant II, so sin2π3=32.

3. 13π4 is not between 0 and 2π, so begin by finding a coterminal angle between 0 and 2π.

   13π4-2π=13π 4-8π4=5π4

   Find the reference angle. 5π4 is in quadrant III and closest x-axis angle is π.

   5π4-π=5π4-4π4=π4

   Evaluate the function of the reference angle using special right triangles or the unit circle.

   tanπ4=1

   13π4 which is coterminal with 5π4 is in quadrant III and tangent is posiive in quandrant III, so tan13π4=1.

4. -7π4 is not between 0 and 2π, so begin by finding a coterminal angle between 0 and 2π.

   - 7π4+2π=-7π4+8π4=π4

   Find the reference angle. π4 is in quadrant I and closest x-axis angle is 0.

   π4-0=π4

   Evaluate the function of the reference angle using special right triangles or the unit circle.

   sinπ4= 22

   -7π4 which is coterminal with π4 is in quadrant I and sine is posiive in quandrant I, so sin- 7π4=22.

Try It 5

Evaluate cos5π3.

½

Example 6: Evaluate Trigonometric Functions

If sinθ=-23 and θ terminates in quadrant IV, find tan θ.

One method to solve this problem is to sketch a right triangle in the specified quadrant with an acute angle at the origin and right angle on the x-axis. The sides of the triangle are from the given trigonometric function. Since sin θ=yr=-23. The r is always positive so r = 3 and y = −2. By using the Pythagorean Theorem, find x.

x2+y2=r2x 2+-22=32x=±5

Since the triangle is in quadrant IV, the x value is positive and x=5

Now evaluate tan θ using the right triangle.

tanθ=yx=-25=-255

Try It 6

If tan α = −2 and α terminates in quadrant II, find sin α.

255

Lesson Summary

where θ is an angle in standard position with point (x, y) on the terminal side and r=x2+y2

The reference angle is the angle between the terminal side of an angle in standard position and the nearest x-axis. Reference angles are always less than π2. The values of the trigonometric functions of the angle in standard position equal values of the trigonometric functions of the reference angle with the appropriate negative signs for the quadrant.

1. Determine the measure of the angle whose terminal side is on the x-axis nearest the terminal side of the given angle.
2. Subtract the measures of the given angle and the x-axis angle.

1. Consider the number to be an angle, θ.
2. If θ is not between 0 and 2π, find a coterminal angle between 0 and 2π by adding or subtracting 2π.
3. Find the reference angle.
4. Evaluate the trigonometric function of the reference angle using special right triangles (lesson 4-04) or the unit circle (lesson 4-02).
5. Apply a negative sign as needed based on the quadrant θ is in.

Practice Exercises

Evaluate the six trigonometric functions based on the given point on the terminal side of an angle in standard position.

1. (3, −5)
2. (−2, −7)

Evaluate the six trigonometric functions of the given angle.

3. π2
4. 2π

Evaluate the function of θ.

5. If sin θ=15 and θ is in quadrant II, find a) cos θ and b) tan θ.
6. If secθ=43 and θ is in quadrant IV, find a) sin θ and b) csc θ.
7. If tanθ=-3 4 and sin θ > 0, find a) cos θ and b) sec θ.
8. If cosθ=-817 and tan θ < 0, find a) sin θ and b) cot θ.

Find the reference angle of the given angle.

9. 6π5
10. 4π7
11. -8π9
12. 15π4

Evaluate the given trigonometric functions using reference angles.

13. sin3π4
14. tan11π6
15. cos5π4

Mixed Review

16. (4-04) Let θ be an acute angle. Use the given function value with trigonometric identities to evaluate the given function. If csc θ = 2, find a) cot θ and b) sin θ.
17. (4-04) A student is standing on the third floor of a building 30 feet above the ground. There are two kids on a lawn playing catch with a frisbee. The angles of depression from the student in the building to the kids are 45° and 55°. How far apart are the students?
18. (4-03) Use special right triangles to evaluate a) sinπ3 and b) cotπ4.
19. (4-02) Use the unit circle to evaluate a) cosπ6 and b) tan7π6.
20. (4-01) a) Draw the 17π6 in standard position and find a b) positive and c) negative coterminal angle.

Answers

1. sinθ =-53434,cosθ=33434,tanθ=-53,cscθ=-345,secθ=343 ,cotθ=-35
2. sinθ=-75353,cosθ=-25353,tanθ=72, cscθ=-537,secθ=-532,cotθ=27
3. sinθ=1,cosθ=0,tanθ=undefined ,cscθ=1,secθ=undefined,cotθ=0
4. sinθ=0,cosθ=1,tanθ=0,cscθ=undefined,secθ= 1,cotθ=undefined
5. -245; -2424
6. -74; -477
7. -45; -54
8. 1517 ; -815
9. π5
10. 3π7
11. π9
12. π4
13. 22
14. -33
15. - 22
16. 3; ½
17. about 9 ft
18. 32; 1
19. 32; 33
20. 
    ; 5π6; -7π6

What are the six trigonometric functions of θ?

What are the 6 trigonometric functions calculator?