Trigonometry Mixed Review: Part 1 (4)

Question 1 of 8

Solve for

$theta$

Round your answer to the nearest minute

$theta=$ (35) $°$ (55) $'$

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Sin Ratio

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

Cos Ratio

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

Tan Ratio

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

First we need to identify which trig ratio to use.

One of the known lengths $(29)$ is $\text(adjacent)$ to

$theta$ and the other length $(21)$ is $\text(opposite)$ to

$theta$

Hence, we can use the

$tan \text(ratio)$ to solve for

$theta$

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

$tan theta$	$=$	$\frac{\color{#004ec4}{21}}{\color{#00880a}{29}}$	Plug in the values

$tan theta$	$=$	$0.724$

Use the inverse function for

$tan$ on your calculator to get

$theta$ by itself

$theta$	$=$	$tan^(-1) (0.724)$	The inverse of $tan$ is $tan^(-1)$
$theta$	$=$	$35.9097$	Use the $\text(shift) tan$ function on your calculator
$theta$	$=$	$35°54’35”$	Use the $\text(degrees)$ function on your calculator
$theta$	$=$	$35°55’$	Rounded to the nearest minute

$theta=35°55’$

Question 2 of 8

Solve for

$x$

Round your answer to one decimal place

$x =$ (25.3)

Incorrect

Sin Ratio

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

Cos Ratio

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

Tan Ratio

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

First we need to identify which trig ratio to use.

One of the known angles $(56°)$ has $21$ as an $\text(opposite)$ side and $x$ is the $\text(hypotenuse)$

Hence, we can use the

$sin \text(ratio)$ to solve for

$x$

$sin theta$	$=$	$\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$	$sin \text(ratio)$

$sin (56°)$	$=$	$\frac{\color{#004ec4}{21}}{\color{#e85e00}{x}}$	Plug in the values

Get

$x$ by itself to find its value

$sin (56°)$	$=$	$21/x$

$x xx sin (56°)$	$=$	$21$	Multiply both sides by $x$

$x$	$=$	$21/sin (56°)$	Divide both sides by $sin(56°)$

$x$	$=$	$21/0.8290375726$	Evaluate $sin(56°)$ on the calculator

$x$	$=$	$25.3$	Round to one decimal place

$x=25.3$

Question 3 of 8

Solve for

$theta$

Round your answer to the nearest degree

$theta=$ (73) $°$

Incorrect

Sin Ratio

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

Cos Ratio

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

Tan Ratio

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

First we need to identify which trig ratio to use.

One of the known lengths $(8)$ is $\text(adjacent)$ to

$theta$ and the other length $(28)$ is the $\text(hypotenuse)$

Hence, we can use the

$cos \text(ratio)$ to solve for

$theta$

$cos theta$	$=$	$\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$	$cos \text(ratio)$

$cos theta$	$=$	$\frac{\color{#00880a}{8}}{\color{#e85e00}{28}}$	Plug in the values

$cos theta$	$=$	$0.286$

Use the inverse function for

$cos$ on your calculator to get

$theta$ by itself

$theta$	$=$	$cos^(-1) (0.286)$	The inverse of $cos$ is $cos^(-1)$
$theta$	$=$	$73.381°$	Use the $\text(shift) cos$ function on your calculator
$theta$	$=$	$73°$	Rounded to the nearest degree

$theta=73°$

Question 4 of 8

Find the length of

$d$

The given measurements are in metres

Round your answer to the nearest whole number

$d=$ (139) m

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The Angles of Elevation and Depression are the angles created by the upward or downward slope of the hypotenuse.

The lines on top and bottom of the building are parallel, and the hypotenuse that cuts through it creates an angle of depression measured

$47°10’$

Since the angle of elevation (

$theta$ ) is opposite of the angle of depression
(

$47°10’$ ),

$theta$ is also equal to

$47°10’$

Next, we need to identify which trig ratio to use.

Angle $theta$ has

$150 m$ as an $\text(opposite)$ side and

$d$ as an $\text(adjacent)$ side.

Hence, we can use the

$tan \text(ratio)$ to solve for $d$

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

$tan$ $47˚10’$	$=$	$\frac{\color{#004ec4}{150}}{\color{#00880a}{d}}$	Plug in the values

$d xx 1.0786$	$=$	$150$	Cross multiply
$d$	$=$	$150/1.0786$	Divide $1.0786$ from both sides to isolate $d$

$d$	$=$	$139 m$

$d=139 m$

Question 5 of 8

Find the area of the Triangle

The given measurements are in units

Round your answer to one decimal place

$\text(Area )=$ (32.6) $units^2$

Question 6 of 8

Solve for

$theta$

Round your answer to the nearest minute

1. $22˚ 24'$
2. $47˚ 15'$
3. $42˚ 45'$
4. $67˚ 36'$

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Sin Ratio

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

Cos Ratio

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

Tan Ratio

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

First we need to identify which trig ratio to use.

One of the known lengths $(4.9)$ is $\text(opposite)$ to

$theta$ , and the other length $(10.6)$ is $\text(adjacent)$ to

$theta$ , but we only need half of that length to form a right triangle.

$\text(adjacent)$	$=$	$10.6divide2$
$\text(adjacent)$	$=$	$5.3$

Hence, we can use the

$tan \text(ratio)$ to solve for

$theta$

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

$tan theta$	$=$	$\frac{\color{#004ec4}{4.9}}{\color{#00880a}{5.3}}$	Plug in the values

$tan theta$	$=$	$0.9243$

Use the inverse function for

$tan$ on your calculator to get

$theta$ by itself

$theta$	$=$	$tan^(-1) (0.9243)$	The inverse of $tan$ is $tan^(-1)$
$theta$	$=$	$42.7543$	Use the $\text(shift) tan$ function on your calculator
$theta$	$=$	$42°45’15”$	Use the $\text(degrees)$ function on your calculator
$theta$	$=$	$42°45’$	Rounded to the nearest minute

$theta=42°45’$

Question 7 of 8

Find the area of the Triangle

The given measurements are in units

Round your answer to the nearest whole number

$\text(Area )=$ (47) $units^2$

Question 8 of 8

Find the area of the Triangle

The given measurements are in units

Round your answer to the nearest whole number

$\text(Area )=$ (92) $units^2$

Incorrect

Area of a Triangle Formula

$\text(Area )=1/2 xx$ $b$

$times$ $c$

$times sin$ $A$

Cosine Rule (finding an angle)

$cos\color{#004ec4}{A}=\frac{\color{#00880a}{b^2}+\color{#e85e00}{c^2}-\color{#004ec4}{a^2}}{2\color{#00880a}{b}\color{#e85e00}{c}}$

Remember

Uppercase letters represent angles in the triangle
Lowercase letters represent the side lengths

Labelling the triangle

First, we need to find an angle to use for the Area of a Triangle formula

We can use the Cosine Rule (finding an angle) to solve for $A$

$cos$ $A$	$=$	$\frac{\color{#00880a}{b^2}+\color{#e85e00}{c^2}-\color{#004ec4}{a^2}}{2\color{#00880a}{b}\color{#e85e00}{c}}$	Cosine Rule Formula

$cos$ $A$	$=$	$\frac{\color{#00880a}{13.9^2}+\color{#e85e00}{14^2}-\color{#004ec4}{16^2}}{2\color{#00880a}{(13.9)}\color{#e85e00}{(14)}}$	Plug in known values

$cos$ $A$	$=$	$(193.21+196-256)/(389.2)$	Evaluate

$cos$ $A$	$=$	$0.342$

Use the inverse function for

$cos$ on your calculator to get $A$ by itself

$A$	$=$	$cos^-1(0.342)$	The inverse of $cos$ is $cos^-1$
$A$	$=$	$71.094$	Use the shift $cos$ function on your calculator
$A$	$=$	$71.1°$	Rounded to one decimal place

Finally, solve for the area using the Area of a Triangle formula

$A$	$=$	$1/2 xx$ $b$ $times$ $c$ $times sin$ $A$	Area of a Triangle formula

	$=$	$1/2 xx$ $13.9$ $times$ $14$ $times sin$ $71.1°$	Plug in the known lengths

	$=$	$92 units^2$	Rounded to one decimal place

The given measurements are in units, so the area is measured as square units

$\text(Area)=92 units^2$

$a/sinA$	$=$	$c/sinC$	Sine Rule Formula

$14/(sin77°)$	$=$	$7/sinC$	Plug in the values

$sin$ $C$ $xx 14$	$=$	$7 xx sin77°$	Cross multiply

$sin$ $C$	$=$	$(7 xx sin77°)/14$	Divide $14$ from each side to isolate $sinC$

$sin$ $C$	$=$	$6.821/14$	Simplify

$sin$ $C$	$=$	$0.487$

$C$	$=$	$sin^-1(0.487)$	The inverse of $sin$ is $sin^-1$
$C$	$=$	$29.155$	Use the shift $sin$ function on your calculator
$C$	$=$	$29.2°$	Rounded to one decimal place

$B$	$=$	$180°-($ $A$ $+$ $C$ $)$
$B$	$=$	$180°-($ $77$ $+$ $29.2$ $)$	Plug in the known values
$B$	$=$	$73.8°$

$A$	$=$	$1/2 xx$ $b$ $times$ $c$ $times sin$ $A$	Area of a Triangle formula

	$=$	$1/2 xx$ $11.5$ $times$ $7$ $times sin$ $126°$	Plug in the known lengths

	$=$	$32.6 units^2$	Rounded to one decimal place

$\text(Area)$	$=$	$1/2 xx$ $a$ $times$ $c$ $times sin$ $B$	Area of a Triangle formula

	$=$	$1/2 xx$ $14$ $times$ $7$ $times sin$ $73.8°$	Plug in the known lengths

	$=$	$47.0 units^2$	Rounded to one decimal place

Trigonometry Mixed Review: Part 1 (4)

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Quiz summary

Information

1. Question

Hint

Great Work!

Sin Ratio

Cos Ratio

Tan Ratio

2. Question

Excellent!

Sin Ratio

Cos Ratio

Tan Ratio

3. Question

Keep Going!

Sin Ratio

Cos Ratio

Tan Ratio

4. Question

Hint

Fantastic!

5. Question

Nice Job!

Area of a Triangle Formula

Remember

Labelling the triangle

6. Question

Hint

Exceptional!

Sin Ratio

Cos Ratio

Tan Ratio

7. Question

Correct!

Area of a Triangle Formula

Remember

Labelling the triangle

8. Question

Great Work!

Area of a Triangle Formula

Cosine Rule (finding an angle)

Remember

Labelling the triangle

Quizzes