Trigonometry Word Problems 2

Question 1 of 4

Noah and Amelia are standing

91 m

$91 m$ apart and observe a drone at an angle of elevation of

63 °

$63°$ and

42 °

$42°$ respectively. Find the distance

(N D)

$(ND)$ between Noah

(N)

$(N)$ and the drone

(D)

$(D)$ to the nearest metre.

$ND =$ $\text(ND) =$ (170) $m$ $m$

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Sine Rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ $a$ is the side opposite angle $A$ $A$
$b$ $b$ is the side opposite angle $B$ $B$
$c$ $c$ is the side opposite angle $C$ $C$

When to use the Sine Rule

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

Notice that the scenario creates an obtuse triangle. Let

N D

$ND$ be the distance of the two poles on the ground.

Identify the two other angles in the triangle.

First angle:

Remember that a straight line measures

180 °

$180°$ . Subtract

63 °

$63°$ from

180 °

$180°$ to find the measure of the larger angle.

180 - 63

$180-63$

=

$=$

117 °

$117°$

Second angle:

Remember that the sum of the interior angles in a triangle is

180 °

$180°$ . Subtract

42 °

$42°$ and

117 °

$117°$ from

180 °

$180°$ to find the measure of the smaller angle.

180 - 42 - 117

$180-42-117$

=

$=$

21 °

$21°$

Since

1

$1$ side and

2

$2$ angles are now known, use the Sine Rule to find the missing side.

Side

1 = N D

$1=ND$

Angle

1 = 42 °

$1=42°$

Side

2 = 91 m

$2=91 m$

Angle

2 = 21 °

$2=21°$

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}$	$=$ $=$	$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}$

$\frac{\color{#007DDC}{ND}}{\sin\color{#007DDC}{42°}}$	$=$ $=$	$\frac{\color{#00880A}{91}}{\sin\color{#00880A}{21°}}$	Substitute the values

$N D \times sin 21 °$ $NDtimessin21°$	$=$ $=$	$91 \times sin 42 °$ $91timessin42°$	Cross multiply
$N D \times sin 21 °$ $NDtimessin21°$ $\div sin 21 °$ $dividesin21°$	$=$ $=$	$91 \times sin 42 °$ $91timessin42°$ $\div sin 21 °$ $dividesin21°$	Divide both sides by $sin 21 °$ $sin21°$

$N D$ $ND$	$=$ $=$	$\frac{91 \times sin 42 °}{sin 21 °}$ $(91timessin42°)/(sin21°)$

$N D$ $ND$	$=$ $=$	$\frac{91 \times 0.6691306}{0.3583679}$ $(91times0.6691306)/0.3583679$	Use the calculator to simplify

$N D$ $ND$	$=$ $=$	$169.991164$ $169.991164$
$N D$ $ND$	$=$ $=$	$170 m$ $170 m$	Rounded off to the nearest metre

170 m

$170 m$

Question 2 of 4

Noah and Amelia are standing

91 m

$91 m$ apart and observe a drone at an angle of elevation of

63 °

$63°$ and

42 °

$42°$ respectively. Find the height

(h)

$(h)$ of the drone to the nearest metre.

$h =$ $h=$ (151) $m$ $m$

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Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

sin = \frac{opposite}{hypotenuse}

$\sin=\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

Cos Ratio (CAH)

cos = \frac{adjacent}{hypotenuse}

$\cos=\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$

Tan Ratio (TOA)

tan = \frac{opposite}{adjacent}

$\tan=\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

Calculator Buttons to Use

$sin$ $sin$

=

$=$ Sine function

$cos$ $cos$

=

$=$ Cosine function

$tan$ $tan$

=

$=$ Tangent function

DMS or $° ‘ ‘ ‘$ $° ‘ ‘ ‘$

=

$=$ Degree/Minute/Second

$=$ $=$

=

$=$ Equal function

Notice that the scenario creates two right triangles. Focus on the smaller one and label it in reference to the given angle.

opposite = h

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{h}$

hypotenuse = 170

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{170}$

Since we now have the opposite and hypotenuse values, we can use the

sin

$sin$ ratio to find

h

$h$ .

$sin 63 °$ $sin63°$	$=$ $=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

$sin 63 °$ $sin63°$	$=$ $=$	$\frac{\color{#004ec4}{h}}{\color{#e85e00}{170}}$

$170 \times$ $170xx$ $sin 63 °$ $sin63°$	$=$ $=$	$\frac{h}{170}$ $h/170$ $\times 170$ $xx170$	Multiply both sides by $170$ $170$

$170 sin 63 °$ $170sin63°$	$=$ $=$	$h$ $h$
$h$ $h$	$=$ $=$	$170 sin 63 °$ $170sin63°$

Simplify this further by evaluating

sin 63 °

$sin63°$ using the calculator:

1 .

$1.$ Press $sin$ $sin$

2 .

$2.$ Press

63

$63$

3 .

$3.$ Press $=$ $=$

The result will be:

0.89100652

$0.89100652$

Continue solving for

h

$h$ .

sin 63 ° = 0.89100652

$sin63°=0.89100652$

$h$ $h$	$=$ $=$	$170 sin 63 °$ $170sin63°$
	$=$ $=$	$170 \times 0.89100652$ $170times0.89100652$
	$=$ $=$	$151.47$ $151.47$
	$=$ $=$	$151 m$ $151 m$	Rounded off to the nearest metre

151 m

$151 m$

Question 3 of 4

Two support wires, one

13 m

$13 m$ long and the other

12 m

$12 m$ long, are attached to a pole from a common point

A

$A$ . The longer wire is attached to the top of the pole. The shorter wire is attached

4 m

$4 m$ below the longer wire. Find the measurement of

∠ A B D

$/_ABD$ .

Round your answer to the nearest minute

$∠ A B D$ $/_ABD$ = (66) $°$ $°$ (47) $'$

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Cosine Rule

$\cos\color{#007DDC}{A}=\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Since

$3$ sides of the triangle

$ABC$ are given, we can use the Cosine Rule.

First, label the triangle

$ABC$ according to the Cosine Rule.

Substitute the three known values to the Cosine Rule to find

$B$ or $/_ABD$ .

From labelling the triangle, we know that the known values are those with labels

$a, b$ and

$c$ .

$B=/_ABD$

$a=15 cm$

$b=11 cm$

$c=6 cm$

$\cos\color{#007DDC}{B}$	$=$	$\frac{\color{#00880A}{a}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{b}^2}{2\color{#00880A}{a}\color{#9a00c7}{c}}$

$\cos\color{#007DDC}{B}$	$=$	$\frac{\color{#00880A}{13}^2+\color{#9a00c7}{4}^2-\color{#007DDC}{12}^2}{2(\color{#00880A}{13})(\color{#9a00c7}{4})}$	Substitute the values

$cos B$	$=$	$(169+16-144)/(104)$	Simplify

$cos B$	$=$	$41/104$

$B$	$=$	$cos^(-1) (41/104)$	Get the inverse of the cosine

Simplify this further by evaluating

$cos^(-1) (41/104)$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $cos$

$3.$ Press

$41$

$4.$ Press $divide$

$3.$ Press

$104$

$4.$ Press $=$

The result will be:

$66.78199226°$

Proceed with solving for

$/_ABD$ .

$cos^(-1) (41/104)=66.78199226°$

$B$	$=$	$cos^(-1) (41/104)$
$B$	$=$	$66.78199226°$
$B$	$=$	$66°46’55”$	Press DMS on the calculator
$B$ or $/_ABD$	$=$	$66°47’$	Round off to the nearest minute

$66°47’$

Question 4 of 4

Two support wires, one

$13 m$ long and the other

$12 m$ long, are attached to a pole from a common point

$A$ . The longer wire is attached to the top of the pole. The shorter wire is attached

$4 m$ below the longer wire. Find the height of the pole

$BD$ .

Round your answer to

$1$ decimal place

$BD$ = (5.1) $m$

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