Download Department of Mathematics, University of Wisconsin

Department of Mathematics, University of Wisconsin-Madison
Math 114
Worksheet Sections 6.1-6.3
1. Convert each angle in degrees to radians.
(a) 330◦
Solution: 330◦ = 330◦ *
11π
π
=
radians
180◦
6
(b) −180◦
Solution: −180◦ = −180◦ *
π
= −π radians
180◦
2. Convert each angle in radians to degrees.
(a) −
2π
3
Solution: −
(b) −
2π 180◦
2π
=−
*
= −120◦
3
3
π
3π
2
Solution: −
3π 180◦
3π
=−
*
= −270◦
2
2
π
3. Find the radius r if the length of the arc of a circle of radius r is 6 centimeters and the central angle
θ is 14 radian.
Solution: l = rθ [θ in radians] =⇒ 6 = r ∗
1
=⇒ r = 24 cm
4
4. Find the angle θ if the area of the arc of a circle of radius r is 8 square meters and the radius of the
circle formed by the central angle θ is 6 meters.
Solution: Area =
θ
θ
θ
4
∗ πr2 =⇒ 8 = ∗ 62 =⇒ 8 = ∗ 36 =⇒ θ = radians
2π
2
2
9
5. Find the exact value of each expression. Do not use a calculator.
1
(a) tan 45◦ cos 30◦
√
3
3
=
2
2
√
Solution: tan 45◦ cos 30◦ = 1 ∗
(b) 5 cos 90◦ − 8 sin 270◦
Solution: 5 cos 90◦ − 8 sin 270◦ = 5 ∗ 0 − 8 sin(180◦ + 90◦ ) = 0 − (−8 sin 90◦ ) = 8 ∗ 1 =8
(c) 2 sin π4 + 3 tan 3π
4
√
√
√
1
√ + 3 tan(π − π4 ) = 2 − 3 tan π4 = 2 − 3 ∗ 1 = 2 − 3
Solution: 2 sin π4 + 3 tan 3π
4 = 2∗
2
6. The point (−1, −2) is on the terminal side of an angle θ in standard position. Find the exact value of
each of the six trigonometric functions of θ.
Solution: We are in the third quadrant, so all trigonometric functions except tan θ and cot θ will
be neagtive.
√
√
For the given angle, |Opposite|
= 2, |Adjacent| = 1, and|Hypotenuse| = 22 + 12 = 5
√
sin θ = − √25
csc θ = − 25
√
cos θ = − √15
− sec θ = 5
tan θ = 2
cot θ = 12
7. Find the exact value of each expression. Do not use a calculator.
(a) sin 390◦
Solution: sin 390◦ = sin(360◦ + 30◦ ) = sin 30◦ =
1
2
(b) sec 420◦
Solution: sec 420◦ =
1
1
1
1
=
=
= 1 =2
cos 420◦
cos(360◦ + 60◦ )
cos 60◦
2
(c) cos 270◦
Solution: cos 270◦ = cos(180◦ + 90◦ ) = − cos 90◦ = 0
(d) sin2 40◦ + cos2 40◦
2
Solution: sin2 40◦ + cos2 40◦ = 1
(e) tan 200◦ · cot 20◦
Solution: tan 200◦ · cot 20◦ = tan(180◦ + 20◦ ) · cot 20◦ = tan 20◦ · cot 20◦ = 1
(f) f (−a) if f (θ) = tan θ and f (a) = 2
Solution: f (−a) = tan(−a) = − tan a = −f (a) = −2
(g) f (a) + f (a + π) + f (a + 2π) if f (θ) = tan θ and f (a) = 2
Solution: f (a) + f (a + π) + f (a + 2π) = tan(a) + tan(a + π) + tan(a + 2π) =
tan(a) + tan(a) + tan(a) = 3 tan(a) = 3f (a) = 3 ∗ 2 = 6
8. Find the exact value of each remaining trigonometric functions of θ if sin θ = − 23 , π < θ <
3π
2 .
Solution: We are in the third quadrant, so all trigonometric functions except tan θ and cot θ will
be negative.
p
√
Since sin θ = − 23 , if |Opposite| = 2, then |Hypotenuse| = 3, and |Adjacent| = (32 − 22 ) = 5
csc θ = − 23
√
tan θ = √25
cot θ = 25
√
cos θ = −
5
3
sec θ = − √35
9. Find the exact value of: sin 1◦ + sin 2◦ + sin 3◦ + ... + sin 358◦ + sin 359◦
Solution:
sin 1◦ + sin 2◦ + sin 3◦ + ... + sin 358◦ + sin 359◦
= sin 1◦ + sin 2◦ + .... sin 179◦
+ sin 180◦
◦
◦
◦
+ sin 359
+ sin 358
+ .... sin 181
= sin 1◦ + sin 2◦ + .... sin 179◦
+ sin 180◦
◦
◦
◦
+ sin −1
+ sin −2
+ .... sin −179
= 0 + 0 + ......0 + sin 180◦ = 0
3