Deck 15: Oblique Triangles and Vectors

Question 1

State whether the following expressions are positive or negative, without using your calculator or your book.
(a) $\cos 135^{\circ}$
(b) $\sin 179^{\circ}$
(c) $\cot 147^{\circ}$
(d) $\csc \left(-700^{\circ}\right)$
(e) $\tan \left(439^{\circ}\right)$
(f) $\sec \left(-149^{\circ}\right)$

Answer 1

(a) negative
(b) positive
(c) negative
(d) positive
(e) positive
(f) negative

Question 2

State whether the following expressions are positive or negative, without using your calculator or your book.
(a) $\cos 225^{\circ}$
(b) $\sin 314^{\circ}$
(c) $\cot 495^{\circ}$
(d) $\csc 820^{\circ}$
(e) $\tan \left(-220^{\circ}\right)$
(f) $\sec 195^{\circ}$

Answer 2

(a) negative
(b) negative
(c) negative
(d) positive
(e) negative
(f) negative

Question 3

In which quadrant will the terminal side of $\theta$ lie if $\theta=192^{\circ}$ ?

Answer 3

III

Question 4

In which quadrant will the terminal side of $\theta$ lie if $\theta=235^{\circ}$ ?

Question 5

In which quadrant will the terminal side of $\theta$ lie if $\theta=-235^{\circ}$ ?

Question 6

In which quadrant will the terminal side of $\theta$ lie if $\theta=993^{\circ}$ ?

Question 7

Give the algebraic signs of the sine, cosine, and tangent of $-145^{\circ}$ .

Question 8

Give the algebraic signs of the sine, cosine, and tangent of $315^{\circ}$ .

Question 9

Give the algebraic signs of the sine, cosine, and tangent of $176^{\circ}$ .

Question 10

Write the sine, cosine, and tangent of the angle $\theta$ whose terminal side passes through the point $(6,-5)$ . Use three significant digits in your answers.

Question 11

Write the sine, cosine, and tangent of the angle $\theta$ whose terminal side passes through the point $(-3,4)$ . Use three significant digits in your answers.

Question 12

Write the sine, cosine, and tangent of the angle $\theta$ whose terminal side passes through the point ( -1 , $-5)$ . Use three significant digits in your answers.

Question 13

Write the sine, cosine and tangent of $\theta$ , where $\cos \theta=5 / 13$ and $\sin \theta$ is positive. Leave your answers in fractional or radical form.

Question 14

Write the sine, cosine and tangent of $\theta$ , where $\sin \theta=5 / 13$ and $\tan \theta$ is positive. Leave your answers in fractional or radical form.

Question 15

Write the sine, cosine, and tangent of $\theta$ , where $\sec \theta=-9 / 7$ and $\tan \theta$ is positive. Leave your answers in fractional or radical form.

Question 16

Write the sine, cosine, and tangent of $\theta$ , where $\csc \theta=-3 / 2$ and $\theta$ is in the third quadrant. Leave your answers in fractional or radical form.

Question 17

Write the sine, cosine, and tangent of $\theta$ , where $\tan \theta=-4 / 5$ and $\theta$ is not in the fourth quadrant. Leave your answers in fractional or radical form.

Question 18

Write the sine, cosine, and tangent of $\theta$ , where $\cot \theta=-4 / 5$ and $\theta$ is not in the second quadrant. Leave your answers in fractional or radical form.

Question 19

Write, to four significant digits, the sine, cosine, and tangent of $343^{\circ}$ .

Question 20

Write, to four significant digits, the sine, cosine, and tangent of $102^{\circ}$ .

Question 21

Write, to four significant digits, the sine, cosine, and tangent of $-105^{\circ}$ .

Question 22

Find, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\tan \theta=2.018$ .

Question 23

Find, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\sin \theta=-0.8412$ .

Question 24

Find, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\cos \theta=-0.5982$ .

Question 25

Evaluate, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\theta=\operatorname{arcsec} 12.02$ .

Question 26

Evaluate, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\theta=\tan^{-1} 1.11$ .

Question 27

Evaluate, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\theta=\sin^{-1} 0.59$ .

Question 28

Evaluate, to the nearest tenth of a degree, all positive angles less than $360^{\circ}$ where $\theta=\csc^{-1} 16$ .

Question 29

Find, to the nearest tenth of a degree, all angles less than $360^{\circ}$ whose trigonometric function is given.
(a) $\cos \theta=0.8660$
(b) $\cot \theta=-2$
(c) $\sec \theta=\infty$
(d) $\csc \theta=5 / 4$
(e) $\tan \theta=1.428$
(f) $\sin \theta=0.135$

Question 30

Find, to the nearest tenth of a degree, all angles less than $360^{\circ}$ whose trigonometric function is given.
(a) $\sin \theta=-0.341$
(b) $\cos \theta=0.138$
(c) $\tan \theta=3.75$
(d) $\csc \theta=7 / 4$
(e) $\sec \theta=3 / 2$
(f) $\cot \theta=-3.75$

Question 31

Evaluate $\tan^{2} 17^{\circ}+\cos^{2} 17^{\circ}$ to four significant digits.

Question 32

Evaluate $\left(\sin 187^{\circ}\right)\left(\tan 43^{\circ}\right)$ to four significant digits.

Question 33

Evaluate $\sin^{2} 172^{\circ}-15 \cos^{2} 32^{\circ}$ to four significant digits.

Question 34

The force in Newtons on the skier shown below is given by $F=m g \cos \theta$ where $m$ is the skier's mass in kilograms, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ is the acceleration due to gravity, and $\theta$ is the angle the slope makes with the horizontal. Calculate the value of $F$ if $m=80.0 \mathrm{~kg}$ and $\theta=30.0^{\circ}$ .

Question 35

Fill in the missing parts of this table for triangle $A B C$ .

Question 36

Fill in the missing parts of this table for triangle $A B C$ .

Question 37

Fill in the missing parts of this table for triangle $A B C$ .

Question 38

Fill in the missing parts of this table for triangle $A B C$ .

Question 39

Fill in the missing parts of this table for triangle $A B C$ .

Question 40

Fill in the missing parts of this table for triangle $A B C$ .

Question 41

Fill in the missing parts of this table for triangle $A B C$ .

Question 42

Solve the oblique triangle $A B C$ if $A=47.8^{\circ}, B=22.1^{\circ}$ , and $b=182$ .

Question 43

Solve the oblique triangle $A B C$ if $B=80.9^{\circ}, b=84.0$ , and $a=57.0$ .

Question 44

Solve the oblique triangle $A B C$ if $A=53.3^{\circ}, C=29.4^{\circ}$ , and $c=12.8$ .

Question 45

Solve the oblique triangle $A B C$ if $B=24^{\circ} 34^{\prime}, C=43^{\circ} 12^{\prime}$ , and $b=3.5$ .

Question 46

Solve the oblique triangle $A B C$ if $C=80.9^{\circ}, b=13.0$ , and $c=17.0$ .

Question 47

Solve the oblique triangle $A B C$ if $A=25.1^{\circ}, B=59.0^{\circ}$ , and $b=3.53 \mathrm{~m}$ .

Question 48

Solve the oblique triangle $A B C$ if $B=34.1^{\circ}, C=99.2^{\circ}$ , and $c=1.00 \mathrm{~cm}$ .

Question 49

Fill in the missing parts of this table for triangle $A B C$ .

Question 50

Fill in the missing parts of this table for triangle $A B C$ .

Question 51

Fill in the missing parts of this table for triangle $A B C$ .

Question 52

Fill in the missing parts of this table for triangle $A B C$ .

Question 53

Fill in the missing parts of this table for triangle $A B C$ .

Question 54

Fill in the missing parts of this table for triangle $A B C$ .

Question 55

Solve the oblique triangle $A B C$ if $A=84.9^{\circ}, b=3.45$ , and $c=2.78$ .

Question 56

Solve the oblique triangle $A B C$ if $a=33, b=28$ , and $c=50$ .

Question 57

Solve the oblique triangle $A B C$ if $a=119, b=152$ , and $C=12.4^{\circ}$ .

Question 58

Solve the oblique triangle $A B C$ if $a=5.5, b=2.7$ , and $C=33^{\circ} 40^{\prime}$ .

Question 59

Solve the oblique triangle $A B C$ if $a=7.52, b=8.40$ , and $c=12.1$ . Round answers to the nearest whole number.

Question 60

Solve the oblique triangle $A B C$ if $a=3.5, b=2.0$ , and $C=49^{\circ}$

Question 61

Solve the oblique triangle $A B C$ if $a=1.5, b=2.2$ , and $c=1$ .1.

Question 62

An observer stands $1.12 \mathrm{~km}$ from a launch pad. At 12:00.00, a rocket leaves the pad and heads straight up at a constant velocity. Four seconds later, the observer measures the angle the rocket makes to the horizontal as $19.29^{\circ}$ . Three seconds after that, the angle the observer measures is $31.49^{\circ}$ . What is the speed of the rocket to the nearest metre per second?

Question 63

A student stands a distance from a building and measures the angle to the top of the building to be
$18.8^{\circ}$ . The student then walks $24.0 \mathrm{ft}$ . directly towards the building and measures the angle to be $20.2^{\circ}$ .
(a) What is the height of the building?
(b) How close was the student to the building for the second measurement?

Question 64

The vertical boom of the quarry crane pictured below is $23.4 \mathrm{~m}$ high. The angle that the swinging boom of the crane makes with the vertical boom is $38.2^{\circ}$ and the swinging boom is $20.0 \mathrm{~m}$ long. How long is the length of cable between the vertical boom and the swinging boom?

Question 65

What must the angle between the handles of the grabber pictured below be in order to pick up an object 12.7 cm wide?

Question 66

A $10.2 \mathrm{~m}$ pole and a $9.1 \mathrm{~m}$ pole are attached by a $3.1 \mathrm{~m}$ long string. The other ends of the poles are hinged together. What is the angle $A$ for which the string is taut?

Question 67

Three circles are to be bored in a plate. The circle $A$ with a diameter $4.00 \mathrm{~cm}$ and the circle $B$ with a diameter $5.50 \mathrm{~cm}$ are tangent to the circle $C$ with a diameter $7.50 \mathrm{~cm}$ . Determine the distance between the centre of circle $\mathrm{A}$ and the centre of circle $\mathrm{B}$ .

Question 68

A triangular lot measures $257 \mathrm{~m}, 316 \mathrm{~m}$ , and $177 \mathrm{~m}$ along its sides. Find the angles between the sides.

Question 69

A ship is $76.0 \mathrm{~km}$ directly west of a port. If the ship sails northwest for $14.0 \mathrm{~km}$ , how far will it be from the port?

Question 70

The sides of a triangle are 11, 14, and 19. Find the length of the median drawn to the shortest side.

Question 71

A triangular lot measures $175 \mathrm{~m}, 160 \mathrm{~m}$ , and $200 \mathrm{~m}$ along its sides. Find the angles between the sides to the nearest second.

Question 72

A ship sails due north for 85.4 nautical miles, then $\mathrm{N} 23^{\circ} \mathrm{E}$ for 14.7 nautical miles. How far is it from where it started?

Question 73

Point $A$ is $45.0 \mathrm{~m}$ from point $B$ and $64.5 \mathrm{~m}$ from point $C$ . An observer at $A$ measures angle $B A C$ to be $34.0^{\circ}$ . How far apart are $B$ and $C$ ?

Question 74

A building is going to be $6.1 \mathrm{~m}$ wide. The roof will be supported by two $4.6-\mathrm{m}$ -rafters. What is the angle at the roof peak?

Question 75

A building is going to be $7.10 \mathrm{~m}$ wide. The roof will be supported by two rafters, one $2.25 \mathrm{~m}$ long and the other $5.25 \mathrm{~m}$ long. What is the angle at the roof peak?

Question 76

Two vectors of magnitude $A=16.6$ and $B=19.0$ are separated by an angle $\theta=61.0^{\circ}$ . Find the resultant and the angle the resultant makes with vector $B$ .

Question 77

Two vectors of magnitude $A=346$ and $B=437$ are separated by an angle $\theta=19.4^{\circ}$ . Find the resultant and the angle the resultant makes with vector $B$ .

Question 78

Two forces of $62 \mathrm{~kg}$ and $112 \mathrm{~kg}$ act on an object with an angle of $54^{\circ}$ between their lines of action. Find the magnitude of their resultant and its direction with respect to the larger force.

Question 79

The resultant of two forces of $1500 \mathrm{~N}$ and $2500 \mathrm{~N}$ is $3200 \mathrm{~N}$ . What angle does the resultant force make with each of the forces?

Question 80

Three forces are in equilibrium: 150 N, 300 N, and 250 N. Find the angles between their lines of action.