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Trigonometry Mixed Review: Part 2

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Trigonometry Mixed Review: Part 2 (3)

Trigonometry Mixed Review: Part 2 (3)

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Question 1 of 7

Find the length of

$a$

Round your answer to two decimal places

$a=$ (12.48) $\text(m)$

Incorrect

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Cosine Rule (finding a length)

$a^2$

$=$ $b^2$

$+$ $c^2$

$-2$ $b$ $c$

$xx cos$ $A$

Cosine Rule (finding an angle)

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

Remember

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

Labelling the triangle

We can use the Cosine Rule (finding a length) to find the length of $a$

$a^2$	$=$	$b^2$ $+$ $c^2$ $-2$ $b$ $c$ $xxcos$ $A$	Cosine Rule Formula
$a^2$	$=$	$9.2^2$ $+$ $5.4^2$ $-2$ $(9.2)$ $(5.4)$ $xxcos$ $115°$	Plug in the values
$a^2$	$=$	$84.6+29.16-99.36xxcos115°$	Evaluate
$a^2$	$=$	$155.79$
$sqrt(a^2)$	$=$	$sqrt(155.79)$	Take the square root of both sides
$a$	$=$	$12.48 m$	Rounded to two decimal places

$a=12.48 \text(m)$

Question 2 of 7

Solve for angle

$C$

Round your answer to the nearest minute

$∠C=$ (34) $°$ (3) $'$

Incorrect

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Cosine Rule (finding a length)

$c^2$

$=$ $a^2$

$+$ $b^2$

$-2$ $a$ $b$

$xx cos$ $C$

Cosine Rule (finding an angle)

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

Remember

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

Labelling the triangle

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

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

$cos$ $C$	$=$	$\frac{\color{#004ec4}{5^2}+\color{#00880a}{7^2}-\color{#e85e00}{4^2}}{2\color{#004ec4}{(5)}\color{#00880a}{(7)}}$	Plug in known values

$cos$ $C$	$=$	$(25+49-16)/(70)$	Evaluate

$cos$ $C$	$=$	$0.8286$

Use the inverse function for

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

$C$	$=$	$cos^-1(0.8286)$	The inverse of $cos$ is $cos^-1$
$C$	$=$	$34.0448084$	Use the shift $cos$ function on your calculator
$C$	$=$	$34° 2′ 41.31”$	Use the degrees button on your calculator
$C$	$=$	$34° 3’$	Round up the minutes

$C=34° 3’$

Question 3 of 7

Solve for the following values:

$(i)$ side

$a$

$(ii)$ angle

$B$

$(iii)$ angle

$C$

The given measurements are in units

Round your answer for the angles to nearest decimal degree

$a=$ (34) $\text(units)$
$∠B =$ (26)°
$∠C =$ (58)°

Incorrect

Cosine Rule (finding a length)

$a^2$

$=$ $b^2$

$+$ $c^2$

$-2$ $b$ $c$

$xx cos$ $A$

Cosine Rule (finding an angle)

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

Remember

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

Labelling the triangle

$\text(Solving for) (i) \text(side) a$

We can use the Cosine Rule (finding a length) to find the length of $a$

$a^2$	$=$	$b^2$ $+$ $c^2$ $-2$ $b$ $c$ $xxcos$ $A$	Cosine Rule Formula
$a^2$	$=$	$15^2$ $+$ $29^2$ $-2$ $(15)$ $(29)$ $xxcos$ $96°$	Plug in the values
$a^2$	$=$	$225+841-870xxcos96°$	Evaluate
$a^2$	$=$	$1156.939763$
$sqrt(a^2)$	$=$	$sqrt(1156.939763)$	Take the square root of both sides
$a$	$=$	$34.0 \text(units)$	Rounded to one decimal place

$\text(Solving for) (ii) \text(angle) B$

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

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

$cos$ $B$	$=$	$\frac{\color{#004ec4}{29^2}+\color{#e85e00}{34^2}-\color{#00880a}{15^2}}{2\color{#004ec4}{(29)}\color{#e85e00}{(34)}}$	Plug in known values

$cos$ $B$	$=$	$(841+1156 –224)/(1972)$	Evaluate

$cos$ $B$	$=$	$0.8990872211$

Use the inverse function for

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

$B$	$=$	$cos^-1(0.8990872211)$	The inverse of $cos$ is $cos^-1$
$B$	$=$	$25.9616553$	Use the shift $cos$ function on your calculator
$B$	$=$	$26.0°$	Rounded to one decimal place

$\text(Solving for) (iii) \text(angle) C$

We can subtract the total of the known values of the angles to

$180°$ to find angle $C$

$C$	$=$	$180-($ $96$ $+$ $26$ $)$	Plug in the known values
$C$	$=$	$180-122$	Evaluate
$C$	$=$	$58°$

$a=34 \text(units)$

$∠B=26°$

$∠C=58°$

Question 4 of 7

Solve for angle

$theta$

Round your answer to the nearest minute

$∠theta=$ (112) $°$ (59) $'$

Question 5 of 7

Solve for the following values:

$(i)$ angle

$B$

$(ii)$ angle

$A$

$(iii)$ side

$a$

The given measurements are in units

Round your answer to one decimal degree

$∠B =$ (65.9) $˚$
$∠A =$ (29.9) $˚$
$a =$ (14, 14.0) $\text(units)$

Question 6 of 7

Solve for angle

$B$

Round your answer to the nearest degree

$∠B=$ (39) $°$

Incorrect

Cosine Rule (finding a length)

$a^2$

$=$ $b^2$

$+$ $c^2$

$-2$ $b$ $c$

$xx cos$ $A$

Cosine Rule (finding an angle)

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

Remember

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

Labelling the triangle

First, we find the length of $a$ by using the Cosine Rule (finding a length)

$a^2$	$=$	$b^2$ $+$ $c^2$ $-2$ $b$ $c$ $xxcos$ $A$	Cosine Rule Formula
$a^2$	$=$	$24^2$ $+$ $27^2$ $-2$ $(24)$ $(27)$ $xxcos$ $96°$	Plug in the values
$a^2$	$=$	$576+729-1296xxcos96°$	Evaluate
$a^2$	$=$	$1440.468888$
$sqrt(a^2)$	$=$	$sqrt(1440.468888)$	Take the square root of both sides
$a$	$=$	$37.95$	Rounded to two decimal places

Now, we can use the Cosine Rule (finding an angle) to solve for $B$

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

$cos$ $B$	$=$	$\frac{\color{#004ec4}{27^2}+\color{#e85e00}{37.95^2}-\color{#00880a}{24^2}}{2\color{#004ec4}{(27)}\color{#e85e00}{(37.95)}}$	Plug in known values

$cos$ $B$	$=$	$(729+1440.2025-576)/(2049.3)$	Evaluate

$cos$ $B$	$=$	$0.7774374177$

Use the inverse function for

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

$B$	$=$	$cos^-1(0.7774374177)$	The inverse of $cos$ is $cos^-1$
$B$	$=$	$38.9734571$	Use the shift $cos$ function on your calculator
$B$	$=$	$39°$	Rounded to the nearest whole number

$B=39°$

Question 7 of 7

Solve for angle

$C$

Round your answer to one decimal degree

$∠C=$ (40.8) $°$

Incorrect

Cosine Rule (finding a length)

$b^2$

$=$ $a^2$

$+$ $c^2$

$-2$ $a$ $c$

$xx cos$ $B$

Cosine Rule (finding an angle)

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

Remember

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

Labelling the triangle

First, we find the length of $b$ by using the Cosine Rule (finding a length)

$b^2$	$=$	$a^2$ $+$ $c^2$ $-2$ $a$ $c$ $xx cos$ $B$	Cosine Rule Formula
$b^2$	$=$	$22.3^2$ $+$ $20^2$ $-2$ $(22.3)$ $(20)$ $xx cos$ $92.4°$	Plug in the values
$b^2$	$=$	$497.29+400-892xxcos92.4°$	Evaluate
$b^2$	$=$	$934.6430831$
$sqrt(b^2)$	$=$	$sqrt(934.6430831)$	Take the square root of both sides
$b$	$=$	$30.57$	Rounded to two decimal places

Now, we can use the Cosine Rule (finding an angle) to solve for $C$

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

$cos$ $C$	$=$	$\frac{\color{#004ec4}{22.3^2}+\color{#00880a}{30.57^2}-\color{#e85e00}{20^2}}{2\color{#004ec4}{(22.3)}\color{#00880a}{(30.57)}}$	Plug in known values

$cos$ $C$	$=$	$(497.29+934.5249 –400)/(1363.422)$	Evaluate

$cos$ $C$	$=$	$0.7567832263$

Use the inverse function for

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

$C$	$=$	$cos^-1(0.7567832263)$	The inverse of $cos$ is $cos^-1$
$C$	$=$	$40.81857096$	Use the shift $cos$ function on your calculator
$C$	$=$	$40.8°$	Rounded to one decimal place

$C=40.8°$