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Angles of Elevation and Depression

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Angles of Elevation and Depression

Angles of Elevation and Depression

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Question 1 of 7

From the top of a control tower at an airport, the angle of depression of a plane on a tarmac is

$41°22’$ . The height of the control tower is

$51 m$ . Find the distance between the plane and the central base of the control tower

$(x)$ to the nearest metre.

$x=$ (58) $m$

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Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

Angle Relationships with Parallel Lines

Alternate Angles

Corresponding Angles

Co-Interior Angles

Alternate Angles are equal.

First, knowing that alternate angles are equal, label the angle inside the triangle which is the Angle of Elevation.

Now, label the triangle in reference to the angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{51}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{x}$

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

$tan$ ratio to find

$x$ .

$tan41°22’$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan41°22’$	$=$	$\frac{\color{#004ec4}{51}}{\color{#00880a}{x}}$

$x$	$=$	$51/(tan41°22′)$	Swap the constant on the left side and the denominator on the right side

Simplify this further by evaluating

$tan41°22’$ using the calculator:

$1.$ Press $tan$

$2.$ Press

$41$ and DMS or $° ‘ ‘ ‘$

$3.$ Press

$22$ and DMS or $° ‘ ‘ ‘$ again

$4.$ Press $=$

The result will be:

$0.880585$

Continue solving for

$x$ .

$tan41°22’=0.880585$

$x$	$=$	$51/(tan41°22′)$

	$=$	$51/0.880585$

	$=$	$57.916$
	$=$	$58 m$	Rounded off to the nearest metre

$58 m$

Question 2 of 7

From the point on top of a building that is

$135m$ tall, the angle of depression of a car is

$45°39’$ . What is the distance

$(x)$ of the car from the foot of the building (nearest metre)?

$x=$ (132) $m$

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Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

Angle Relationships with Parallel Lines

Alternate Angles

Corresponding Angles

Co-Interior Angles

Alternate Angles are equal.

First, knowing that alternate angles are equal, label the angle inside the triangle which is the Angle of Elevation.

Now, label the triangle in reference to the angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{135}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{x}$

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

$tan$ ratio to find

$x$ .

$tan45°39’$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan45°39’$	$=$	$\frac{\color{#004ec4}{135}}{\color{#00880a}{x}}$

$x$	$=$	$135/(tan45°39′)$	Swap the constant on the left side and the denominator on the right side

Simplify this further by evaluating

$tan45°39’$ using the calculator:

$1.$ Press $tan$

$2.$ Press

$45$ and DMS or $° ‘ ‘ ‘$

$3.$ Press

$39$ and DMS or $° ‘ ‘ ‘$ again

$4.$ Press $=$

The result will be:

$1.02295$

Continue solving for

$x$ .

$tan45°39’=1.02295$

$x$	$=$	$135/(tan45°39′)$

	$=$	$135/1.02295$

	$=$	$131.971$
	$=$	$132 m$	Rounded off to the nearest metre

$132 m$

Question 3 of 7

A section of a roller coaster ride has a

$57°$ angle of depression. If the length of the section is

$42m$ , what is the length

$(h)$ of its vertical drop (

$1$ decimal place)?

$h=$ (35.2) $m$

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Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, redraw the triangle outside the rollercoaster so that the given angle can be used.

Now, label the triangle in reference to the angle.

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

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{42}$

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

$sin$ ratio to find

$h$ .

$sin57°$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

$sin57°$	$=$	$\frac{\color{#004ec4}{h}}{\color{#e85e00}{42}}$

$42xx$ $sin57°$	$=$	$h/42$ $xx42$	Multiply both sides by $42$

$42sin57°$	$=$	$h$
$h$	$=$	$42sin57°$

Simplify this further by evaluating

$sin57°$ using the calculator:

$1.$ Press $sin$

$2.$ Press

$45$ and DMS or $° ‘ ‘ ‘$

$3.$ Press $=$

The result will be:

$0.83867$

Continue solving for

$h$ .

$sin57°=0.83867$

$h$	$=$	$42sin57°$

	$=$	$42xx0.83867$

	$=$	$35.224$
	$=$	$35.2 m$	Rounded off to $1$ decimal place

$35.2 m$

Question 4 of 7

Two buildings of different heights stand on the opposite sides of the street and are

$39$ m apart. A man standing on the roof of the shorter building reaches an angle of elevation of

$49°58’$ to look at the roof of the taller building. If the shorter building is

$68$ m tall, what is the height

$(h)$ of the taller building to the nearest metre?

$h=$ (114)m

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Mute

Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, form a right triangle from the given diagram and label its values.

Let

$x$ be the difference in height between the two buildings.

Later on, we can solve for

$h$ by adding

$68$ m and the value of

$x$ .

Now, label the triangle in reference to the known angle.

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

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{39}$

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

$tan$ ratio to find

$x$ .

$tan49°58’$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan49°58’$	$=$	$\frac{\color{#004ec4}{x}}{\color{#00880a}{39}}$

$39times$ $tan49°58’$	$=$	$\frac{\color{#004ec4}{x}}{\color{#00880a}{39}}\color{#CC0000}{\times39}$	Multiply both sides by $39$

$39tan49°58’$	$=$	$x$
$x$	$=$	$39tan49°58’$

Simplify this further by evaluating

$tan49°58’$ using the calculator:

$1.$ Press $tan$

$2.$ Press

$49$ and DMS or $° ‘ ‘ ‘$

$3.$ Press

$58$ and DMS or $° ‘ ‘ ‘$ again

$4.$ Press $=$

The result will be:

$1.1903465$

Continue solving for

$x$ .

$tan49°58’=1.1903465$

$x$	$=$	$39tan49°58’$
	$=$	$39times1.1903465$
	$=$	$46.4235$
	$=$	$46$ m	Rounded off to the nearest metre

Finally, solve for

$h$ by adding the value of

$x$ and the height of the shorter building.

$h$	$=$	$x+68$
	$=$	$46+68$
	$=$	$114$ m

$114$ m

Question 5 of 7

Mia finds the angle of elevation of a cliff

$310m$ above ground level to be

$20°$ . After walking towards the cliff, she finds that the angle of elevation increases to

$29°$ . How far did Mia walk

$(x) ?$ (nearest metre)

$x=$ (292) $m$

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Mute

Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

Notice that the scenario creates two right triangles. Subtracting their bases will give us

$x$ .

Label each triangle bases with $a$ and $b$ .

$a$

$-$ $b$

$=x$

Draw each triangle separately and solve for their bases.

Larger triangle with base

$a$ :

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{310}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{a}$

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

$tan$ ratio to find $a$ .

$tan20°$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan20°$	$=$	$\frac{\color{#004ec4}{310}}{\color{#00880a}{a}}$

$a$	$=$	$310/(tan20°)$	Swap the constant on the left side and the denominator on the right side

$a$	$=$	$310/0.36397$	Press $tan$ $20$ $=$ on your calculator

$a$	$=$	$851.719$

Smaller triangle with base

$b$ :

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{310}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{b}$

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

$tan$ ratio to find $b$ .

$tan29°$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan29°$	$=$	$\frac{\color{#004ec4}{310}}{\color{#00880a}{b}}$

$b$	$=$	$310/(tan29°)$	Swap the constant on the left side and the denominator on the right side

$b$	$=$	$310/0.5543$	Press $tan$ $29$ $=$ on your calculator

$b$	$=$	$559.255$

Finally, get the difference of the two bases to solve for

$x$ .

$a=851.719$

$b=559.255$

$x$	$=$	$a$ $-$ $b$
	$=$	$851.719$ $-$ $559.255$
	$=$	$292.464 m$
	$=$	$292 m$	Round off to the nearest metre

$292 m$

Question 6 of 7

A radar station measures the angle of elevation of a missile to be

$31°10’$ and the line of sight distance is

$58 km$ . What is the altitude

$($ height

$,h)$ of the missile? (nearest kilometre)

$h=$ (30) $km$

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Mute

Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the angle.

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

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{58}$

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

$sin$ ratio to find

$h$ .

$sin31°10’$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

$sin31°10’$	$=$	$\frac{\color{#004ec4}{h}}{\color{#e85e00}{58}}$

$58xx$ $sin31°10’$	$=$	$h/58$ $xx58$	Multiply both sides by $58$

$58sin31°10’$	$=$	$h$
$h$	$=$	$58sin31°10’$

Simplify this further by evaluating

$sin31°10’$ using the calculator:

$1.$ Press $sin$

$2.$ Press

$31$ and DMS or $° ‘ ‘ ‘$

$3.$ Press

$10$ and DMS or $° ‘ ‘ ‘$ again

$4.$ Press $=$

The result will be:

$0.517529$

Continue solving for

$h$ .

$sin31°10’=0.517529$

$h$	$=$	$58sin31°10’$
	$=$	$58xx0.517529$
	$=$	$30.0167$
	$=$	$30 km$	Rounded off to the nearest kilometre

$30 km$

Question 7 of 7

A plane is flying at an altitude of

$950 m$ . Emily, who is standing on the ground observes the angle of elevation to the plane at

$70°$ . Then a few seconds later, the angle or elevation has changed to

$25°$ . What is the distance

$(x)$ that the plane flown in these seconds?

$x=$ (1692) $m$

Incorrect

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Mute

Trigonometric Ratios (SOHCAHTOA)

Sin Ratio (SOH)

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

Cos Ratio (CAH)

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

Tan Ratio (TOA)

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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

Angle Relationships with Parallel Lines

Alternate Angles

Corresponding Angles

Co-Interior Angles

Alternate Angles are equal.

First, knowing that alternate angles are equal, use the given Angles of Elevation to form Angles of Depression.

Notice that the scenario creates two right triangles. Subtracting their top sides will give us

$x$ .

Label each of those sides with $a$ and $b$ .

$x=$ $a$

$-$ $b$

Draw each triangle separately and solve for their top sides.

Larger triangle with side

$a$ :

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{950}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{a}$

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

$tan$ ratio to find $a$ .

$tan25°$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan25°$	$=$	$\frac{\color{#004ec4}{950}}{\color{#00880a}{a}}$

$a$	$=$	$950/(tan25°)$	Swap the constant on the left side and the denominator on the right side

$a$	$=$	$950/0.466308$	Press $tan$ $25$ $=$ on your calculator

$a$	$=$	$2037.28$

Smaller triangle with side

$b$ :

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{950}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{b}$

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

$tan$ ratio to find $b$ .

$tan70°$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

$tan70°$	$=$	$\frac{\color{#004ec4}{950}}{\color{#00880a}{b}}$

$b$	$=$	$950/(tan70°)$	Swap the constant on the left side and the denominator on the right side

$b$	$=$	$950/2.747477$	Press $tan$ $70$ $=$ on your calculator

$b$	$=$	$345.771$

Finally, get the difference of the two sides to solve for

$x$ .

$a=2037.28$

$b=345.771$

$x$	$=$	$a$ $-$ $b$
	$=$	$2037.28$ $-$ $345.771$
	$=$	$1691.509 m$
	$=$	$1692 m$	Round off to the nearest metre

$1692 m$