Trigonometry Mixed Review: Part 1 (2)

Question 1 of 9

Solve for

$d$

Round your answer to two decimal places

$d =$ (12.86)

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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 angles $(37°42′)$ has $d$ as an $\text(adjacent)$ side and the other length $(15.5)$ is the $\text(hypotenuse)$

Hence, we can use the

$cos \text(ratio)$ to solve for

$d$

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

$cos (37°42′)$	$=$	$\frac{\color{#00880a}{d}}{\color{#e85e00}{15.5}}$	Plug in the values

Get

$d$ by itself to find its value

$cos (37°42′)$	$=$	$d/15.5$

$15.5 xx cos (65°)$	$=$	$d$	Multiply both sides by $15.5$
$15.5 xx 0.829$	$=$	$d$	Evaluate $cos(37°42′)$ on the calculator
$12.86$	$=$	$d$	Round to two decimal places
$d$	$=$	$12.86$

$d=12.86$

Question 2 of 9

Solve for

$theta$

Round your answer to the nearest degree

$theta=$ (24) $°$

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 $(22)$ is $\text(opposite)$ to

$theta$ and the other length $(55)$ is the $\text(hypotenuse)$

Hence, we can use the

$sin \text(ratio)$ to solve for

$theta$

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

$sin theta$	$=$	$\frac{\color{#004ec4}{22}}{\color{#e85e00}{55}}$	Plug in the values

$sin theta$	$=$	$0.4$

Use the inverse function for

$sin$ on your calculator to get

$theta$ by itself

$theta$	$=$	$sin^(-1) (0.4)$	The inverse of $sin$ is $sin^(-1)$
$theta$	$=$	$23.578°$	Use the $\text(shift) sin$ function on your calculator
$theta$	$=$	$24°$	Rounded to the nearest degree

$theta=24°$

Question 3 of 9

Solve for

$a$

Round your answer to two decimal places

$a =$ (19.37)

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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 angles $(61°15′)$ has $a$ as an $\text(opposite)$ side and $13.5$ as an $\text(adjacent)$ side

Hence, we can use the

$tan \text(ratio)$ to solve for

$a$

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

$tan (61°15′)$	$=$	$\frac{\color{#004ec4}{a}}{\color{#00880a}{13.5}}$	Plug in the values

Now we need to have

$a$ on one side of the equation

$tan (61°15′)$	$=$	$a/13.5$

$13.5 times tan (61°15′)$	$=$	$a$	Multiply both sides by $13.5$
$13.5 times 1.43$	$=$	$a$	Evaluate $tan (61°15′)$ on the calculator
$19.37$	$=$	$a$	Round to two decimal places
$a$	$=$	$19.37$

$a=19.37$

Question 4 of 9

Solve for

$theta$

Round your answer to the nearest minute

1. $49˚ 41'$
2. $40˚ 19'$
3. $39˚ 41'$
4. $35˚ 19'$

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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 $(14)$ is $\text(adjacent)$ to

$theta$ and the other length $(16.5)$ is $\text(opposite)$ to

$theta$

Hence, we can use the

$tan \text(ratio)$ to solve for

$theta$

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

$tan theta$	$=$	$\frac{\color{#004ec4}{16.5}}{\color{#00880a}{14}}$	Plug in the values

$tan theta$	$=$	$1.1786$

Use the inverse function for

$tan$ on your calculator to get

$theta$ by itself

$theta$	$=$	$tan^(-1) (1.1786)$	The inverse of $tan$ is $tan^(-1)$
$theta$	$=$	$49.686$	Use the $\text(shift) tan$ function on your calculator
$theta$	$=$	$49°41’9”$	Use the $\text(degrees)$ function on your calculator
$theta$	$=$	$49°41’$	Rounded to the nearest minute

$theta=49°41’$

Question 5 of 9

Find the length of

$x$

Round your answer to one decimal place

$x=$ (69.3) m

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

First, we need to label the components of the triangle.

To solve for

$x$ , we need to subtract the value of $b$ from the value of $a$

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

The

$28°$ angle has $a$ as an $\text(adjacent)$ side, the

$54°$ angle has $b$ as an $\text(adjacent)$ side, and both angles have

$60 m$ as their $\text(opposite)$ side.

Hence, we can use the

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

$a$ and

$b$

Solve for the value of $a$ first:


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

$tan28°$	$=$	$\frac{\color{#004ec4}{60}}{\color{#cc0000}{a}}$	Plug in the values

$a$ $xx tan28°$	$=$	$60$	Cross multiply
$a$	$=$	$60/(tan28°)$	Divide $tan28°$ from both sides to isolate $a$

$a$	$=$	$112.8 m$	Rounded to one decimal place

Next, use the

$tan \text(ratio)$ to solve for

$b$

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

$tan54°$	$=$	$\frac{\color{#004ec4}{60}}{\color{#9e8600}{b}}$	Plug in the values

$b$ $xx tan54°$	$=$	$60$	Cross multiply
$b$	$=$	$60/(tan54°)$	Divide $tan54°$ from both sides to isolate $b$

$b$	$=$	$43.6 m$	Rounded to one decimal place

Finally, subtract the value of $b$ from the value of $a$ to find

$x$

$x$	$=$	$a$ $-$ $b$
$x$	$=$	$112.8$ $-$ $43.6$	Plug in the values
$x$	$=$	$69.3 m$

$x=69.3 m$

Question 6 of 9

Solve for

$x$

Round your answer to one decimal place

$x =$ (11.7)

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 $(43°)$ has $x$ as an $\text(adjacent)$ side and the other length $(16)$ is the $\text(hypotenuse)$

Hence, we can use the

$cos \text(ratio)$ to solve for

$x$

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

$cos (43°)$	$=$	$\frac{\color{#00880a}{x}}{\color{#e85e00}{16}}$	Plug in the values

Get

$x$ by itself to find its value

$cos (43°)$	$=$	$x/16$

$16 xx cos (43°)$	$=$	$x$	Multiply both sides by $16$

$16 xx 0.7313537016$	$=$	$x$	Evaluate $cos(43°)$ on the calculator

$11.7$	$=$	$x$	Round to one decimal place
$x$	$=$	$11.7$

$x=11.7$

Question 7 of 9

Solve for

$h$

Round your answer to the nearest metre

$h=$ (315, 316) m

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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 angles $(32°15′)$ has

$h$ as an $\text(opposite)$ side and

$500$ as an $\text(adjacent)$ side

Hence, we can use the

$tan \text(ratio)$ to solve for

$x$

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

$tan (32°15′)$	$=$	$\frac{\color{#004ec4}{h}}{\color{#00880a}{500}}$	Plug in the values

Now we need to have

$x$ on one side of the equation

$tan (32°15′)$	$=$	$h/500$

$500 times tan (32°15′)$	$=$	$h$	Multiply both sides by $500$
$500 times 0.631$	$=$	$h$	Evaluate $cos(43°)$ on the calculator
$315$	$=$	$h$	Rounded to the nearest metre
$h$	$=$	$315 m$

$h=315 m$

Question 8 of 9

Solve for

$theta$

Round your answer to the nearest minute

$theta=$ (69) $°$ (20) $'$

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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 $(6)$ is $\text(adjacent)$ to

$theta$ and the other length $(17)$ is the $\text(hypotenuse)$

Hence, we can use the

$cos \text(ratio)$ to solve for

$theta$

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

$cos theta$	$=$	$\frac{\color{#00880a}{6}}{\color{#e85e00}{17}}$	Plug in the values

$cos theta$	$=$	$0.353$	Evaluate $6/17$ to 3 decimal places

Use the inverse function for

$cos$ on your calculator to get

$theta$ by itself

$theta$	$=$	$cos^(-1) (0.353)$	The inverse of $cos$ is $cos^(-1)$
$theta$	$=$	$69.3327$	Use the $\text(shift) cos$ function on your calculator
$theta$	$=$	$69°19’57”$	Use the $\text(degrees)$ function on your calculator
$theta$	$=$	$69°20’$	Round up the minutes

$theta=69°20’$

Question 9 of 9

Solve for angle

$theta$

Round your answer to the nearest minute

1. $22° 21 ’$
2. $17° 6'$
3. $13° 30 ’$
4. $34° 45 ’$

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

First, we need to label the components of the triangle.

To solve for

$theta$ , we need to subtract the added value of $gamma$ and $beta$ from the total interior angle of a triangle, which is

$180°$

In order to find $gamma$ , we need to first solve for $alpha$ .

Angle $alpha$ has

$8.5 m$ as an $\text(opposite)$ side and

$10.2 m$ as an $\text(adjacent)$ side.

Hence, we can use the

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

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

$tan$ $alpha$	$=$	$\frac{\color{#004ec4}{8.5}}{\color{#00880a}{10.2}}$	Plug in the values

$tan$ $alpha$	$=$	$0.833$	Evaluate

Use the inverse function for

$tan$ on your calculator to get $alpha$ by itself

$alpha$	$=$	$tan^(-1) (0.833)$	The inverse of $tan$ is $tan^(-1)$
$alpha$	$=$	$39.794$	Use the $\text(shift) tan$ function on your calculator
$alpha$	$=$	$39° 48′ 20.06″$	Use the $\text(degrees)$ function on your calculator
$alpha$	$=$	$39° 48’$	Rounded to the nearest minute

Recall that a straight line has an angle of

$180°$

Since we have the value of $alpha$ , we can subtract that to

$180°$ to find the value of $gamma$

$gamma$	$=$	$180°-$ $alpha$
$gamma$	$=$	$180°-$ $39°48’$	Plug in the values
$gamma$	$=$	$140° 12’$

Next, identify which trig ratio to use for finding $beta$ .

Angle $beta$ has

$8.5 m$ as an $\text(opposite)$ side and

$17.2 m$ (

$10.2+7$ ) as an $\text(adjacent)$ side.

Thus, use the

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

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

$tan$ $beta$	$=$	$\frac{\color{#004ec4}{8.5}}{\color{#00880a}{17.2}}$	Plug in the values

$tan$ $beta$	$=$	$0.494$	Evaluate

Use the inverse function for

$tan$ on your calculator to get $beta$ by itself

$beta$	$=$	$tan^(-1) (0.494)$	The inverse of $tan$ is $tan^(-1)$
$beta$	$=$	$26.298$	Use the $\text(shift) tan$ function on your calculator
$beta$	$=$	$26° 17′ 53″$	Use the $\text(degrees)$ function on your calculator
$beta$	$=$	$26° 18’$	Rounded to the nearest minute

Finally, we can subtract the added value of $gamma$ and $beta$ from

$180°$ to find the value of

$theta$

$theta$	$=$	$180°-($ $gamma$ $+$ $beta$ $)$
$theta$	$=$	$180°-($ $140°12’$ $+$ $26°18’$ $)$	Plug in the values
$theta$	$=$	$13° 30’$

$theta=13° 30’$

Trigonometry Mixed Review: Part 1 (2)

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Quiz summary

Information

1. Question

Hint

Great Work!

Sin Ratio

Cos Ratio

Tan Ratio

2. Question

Well Done!

Sin Ratio

Cos Ratio

Tan Ratio

3. Question

Hint

Nice Job!

Sin Ratio

Cos Ratio

Tan Ratio

4. Question

Hint

Correct!

Sin Ratio

Cos Ratio

Tan Ratio

5. Question

Hint

Well Done!

6. Question

Correct!

Sin Ratio

Cos Ratio

Tan Ratio

7. Question

Hint

Well Done!

Sin Ratio

Cos Ratio

Tan Ratio

8. Question

Hint

Excellent!

Sin Ratio

Cos Ratio

Tan Ratio

9. Question

Hint

Great Work!

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