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Trigonometry Foundations

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Trig Ratios: Solving for an Angle

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Trig Ratios: Solving for an Angle

Trig Ratios: Solving for an Angle

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Question 1 of 7

Find

$theta$ .

Round your answer to the nearest minute

$theta=$ (67) $°$ (36) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$theta$ .

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{8}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{21}$

Since we now have the adjacent value and the hypotenuse, we can use the

$cos$ ratio to find

$theta$ .

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

$cos theta$	$=$	$\frac{\color{#00880a}{8}}{\color{#e85e00}{21}}$

$theta$	$=$	$cos^-1(8/21)$	Get the inverse of $cos$

Now, follow these steps to evaluate

$cos^-1(8/21)$ using your calculator.

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

$2.$ Press $cos$

$3.$ Press

$8$

$3.$ Press

$÷$

$3.$ Press

$21$

$4.$ Press $=$

This will give a result of

$67.6075°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$67.6075°=67°36’$ $26”$

Since

$26”$ is less than $30”$ , we can round the minute down to

$67°36’$ .

$theta=67°36’$

Question 2 of 7

Find

$theta$ .

Round your answer to the nearest minute

$theta=$ (75) $°$ (15) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$theta$ .

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{19}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{5}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{19}}{\color{#00880a}{5}}$

$theta$	$=$	$tan^-1(19/5)$	Get the inverse of $tan$

Now, follow these steps to evaluate

$tan^-1(19/5)$ using your calculator.

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

$2.$ Press $tan$

$3.$ Press

$19$

$3.$ Press

$÷$

$3.$ Press

$5$

$4.$ Press $=$

This will give a result of

$75.2564°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$75.2564°=75°15’$ $23”$

Since

$23”$ is less than $30”$ , we can round the minute down to

$75°15’$ .

$theta=75°15’$

Question 3 of 7

Find

$theta$ .

Round your answer to the nearest minute

$theta=$ (57) $°$ (48) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$theta$ .

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{22}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{26}$

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

$sin$ ratio to find

$theta$ .

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

$sin theta$	$=$	$\frac{\color{#004ec4}{22}}{\color{#e85e00}{26}}$

$theta$	$=$	$sin^-1(22/26)$	Get the inverse of $sin$

Now, follow these steps to evaluate

$sin^-1(22/26)$ using your calculator.

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

$2.$ Press $sin$

$3.$ Press

$22$

$3.$ Press

$÷$

$3.$ Press

$26$

$4.$ Press $=$

This will give a result of

$57.795°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$57.795°=57°47’$ $43”$

Since

$43”$ is more than $30”$ , we can round the minute up to

$57°48’$ .

$theta=57°48’$

Question 4 of 7

Find

$theta$ .

Round your answer to the nearest minute

$theta=$ (22) $°$ (27) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$theta$ .

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{38}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{92}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{38}}{\color{#00880a}{92}}$

$theta$	$=$	$tan^-1(38/92)$	Get the inverse of $tan$

Now, follow these steps to evaluate

$tan^-1(38/92)$ using your calculator.

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

$2.$ Press $tan$

$3.$ Press

$38$

$3.$ Press

$÷$

$3.$ Press

$92$

$4.$ Press $=$

This will give a result of

$22.49258°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$22.49258°=22°26’$ $33”$

Since

$33”$ is more than $30”$ , we can round the minute up to

$22°27’$ .

$theta=22°27’$

Question 5 of 7

Find

$theta$ .

Round your answer to the nearest minute

$theta=$ (28) $°$ (15) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$theta$ .

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{37}$

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

Since we now have the adjacent value and the hypotenuse, we can use the

$cos$ ratio to find

$theta$ .

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

$cos theta$	$=$	$\frac{\color{#00880a}{37}}{\color{#e85e00}{42}}$

$theta$	$=$	$cos^-1(37/42)$	Get the inverse of $cos$

Now, follow these steps to evaluate

$cos^-1(37/42)$ using your calculator.

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

$2.$ Press $cos$

$3.$ Press

$37$

$3.$ Press

$÷$

$3.$ Press

$42$

$4.$ Press $=$

This will give a result of

$28.242537°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$28.242537°=28°14’$ $33”$

Since

$33”$ is more than $30”$ , we can round the minute up to

$28°15’$ .

$theta=28°15’$

Question 6 of 7

Find

$alpha$ .

Round your answer to the nearest minute

$alpha=$ (53) $°$ (2) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle in reference to

$alpha$ .

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{15.1}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{18.9}$

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

$sin$ ratio to find

$alpha$ .

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

$sin alpha$	$=$	$\frac{\color{#004ec4}{15.1}}{\color{#e85e00}{18.9}}$

$alpha$	$=$	$sin^-1(15.1/18.9)$	Get the inverse of $sin$

Now, follow these steps to evaluate

$sin^-1(15.1/18.9)$ using your calculator.

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

$2.$ Press $sin$

$3.$ Press

$15.1$

$3.$ Press

$÷$

$3.$ Press

$18.9$

$4.$ Press $=$

This will give a result of

$53.02917°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$53.02917°=53°1’$ $45”$

Since

$45”$ is more than $30”$ , we can round the minute up to

$53°2’$ .

$alpha=53°2’$

Question 7 of 7

A boat is about to be launched down a

$21$ -meter ramp with a vertical drop-off of

$7$ m. At what angle (

$theta$ ) is the ramp inclined with the vertical drop-off?

Round your answer to the nearest minute

$theta=$ (70) $°$ (32) $'$

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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

Shift or 2nd F or INV

$=$ Inverse function

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

$=$

$=$ Equal function

First, label the triangle diagram in reference to

$theta$ .

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{7}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{21}$

Since we now have the adjacent value and the hypotenuse, we can use the

$cos$ ratio to find

$theta$ .

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

$cos theta$	$=$	$\frac{\color{#00880a}{7}}{\color{#e85e00}{21}}$

$theta$	$=$	$cos^-1(7/21)$	Get the inverse of $cos$

Now, follow these steps to evaluate

$cos^-1(7/21)$ using your calculator.

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

$2.$ Press $cos$

$3.$ Press

$7$

$3.$ Press

$÷$

$3.$ Press

$21$

$4.$ Press $=$

This will give a result of

$70.528779°$ .

To round off the result to the nearest minute, press the DMS button and check the seconds value.

$70.528779°=70°31’$ $43”$

Since

$43”$ is more than $30”$ , we can round the minute up to

$70°32’$ .

The ramp is inclined

$70°32’$ to the vertical drop-off.

$theta=70°32’$