Trig Ratios: Solving for an Angle

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Question 1 of 7

1. Question

Find `theta`.

Round your answer to the nearest minute

`theta=` (67)`°` (36)`'`

Hint

Help Video

Correct

Awesome!

Incorrect

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

2. Question

Find `theta`.

Round your answer to the nearest minute

`theta=` (75)`°` (15)`'`

Hint

Help Video

Correct

Nice Work!

Incorrect

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

3. Question

Find `theta`.

Round your answer to the nearest minute

`theta=` (57)`°` (48)`'`

Hint

Help Video

Correct

Fantastic!

Incorrect

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

4. Question

Find `theta`.

Round your answer to the nearest minute

`theta=` (22)`°` (27)`'`

Hint

Help Video

Correct

Good Job!

Incorrect

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

5. Question

Find `theta`.

Round your answer to the nearest minute

`theta=` (28)`°` (15)`'`

Hint

Help Video

Correct

Fantastic!

Incorrect

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

6. Question

Find `alpha`.

Round your answer to the nearest minute

`alpha=` (53)`°` (2)`'`

Hint

Help Video

Correct

Perfect!

Incorrect

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

7. Question

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)`'`

Hint

Help Video

Correct

Exceptional!

Incorrect

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’`

Quizzes

Intro to Trigonometric Ratios (SOH CAH TOA) 1
Intro to Trigonometric Ratios (SOH CAH TOA) 2
Round Angles (Degrees, Minutes, Seconds)
Evaluate Trig Expressions using a Calculator 1
Evaluate Trig Expressions using a Calculator 2
Trig Ratios: Solving for a Side 1
Trig Ratios: Solving for a Side 2
Trig Ratios: Solving for an Angle
Angles of Elevation and Depression
Trig Ratios Word Problems: Solving for a Side
Trig Ratios Word Problems: Solving for an Angle
Area of Non-Right Angled Triangles 1
Area of Non-Right Angled Triangles 2
Sine Rule: Solving for a Side
Sine Rule: Solving for an Angle
Cosine Rule: Solving for a Side
Cosine Rule: Solving for an Angle
Trigonometry Word Problems 1
Trigonometry Word Problems 2
Trigonometry Mixed Review: Part 1 (1)
Trigonometry Mixed Review: Part 1 (2)
Trigonometry Mixed Review: Part 1 (3)
Trigonometry Mixed Review: Part 1 (4)
Trigonometry Mixed Review: Part 2 (1)
Trigonometry Mixed Review: Part 2 (2)
Trigonometry Mixed Review: Part 2 (3)