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Trigonometry Foundations

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Trig Ratios Word Problems: Solving for an Angle

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Trig Ratios Word Problems: Solving for an Angle

Trig Ratios Word Problems: Solving for an Angle

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Question 1 of 6

A

$7m$ long ladder is placed against the wall. The ladder is

$5m$ up the wall. What is the angle,

$theta$ , between the ground and the ladder to the nearest minute?

$theta=$ (45) $°$ (35) $'$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the missing angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{5}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{7}$

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

$sin$ ratio to find

$theta$ .

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

$sin theta$	$=$	$\frac{\color{#004ec4}{5}}{\color{#e85e00}{7}}$

$sin theta$	$=$	$0.714286$
$theta$	$=$	$sin^(-1) 0.714286$	Get the inverse of $sin$

Simplify this further by evaluating

$sin^(-1) 0.714286$ using the calculator:

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

$2.$ Press $sin$

$3.$ Press

$0.714286$

$4.$ Press $=$

The result will be:

$45.5847°$

Finally, round off the answer to the nearest minute.

$theta$	$=$	$45.5847°$
	$=$	$45°35’4.9”$	Press DMS on your calculator
	$=$	$45°35’$	Round down since the seconds is less than $30”$

$45°35’$

Question 2 of 6

A cat observes a bird’s nest on top of a tree which is

$18.5 m$ tall. The cat is

$9.5 m$ from the base of the tree. At what angle

$(theta)$ must the cat look up in order to see the nest? (nearest minute)

$theta=$ (62) $°$ (49) $'$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the missing angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{18.5}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{9.5}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{18.5}}{\color{#00880a}{9.5}}$

$tan theta$	$=$	$1.947368$
$theta$	$=$	$tan^(-1) 1.947368$	Get the inverse of $tan$

Simplify this further by evaluating

$tan^(-1) 1.947368$ using the calculator:

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

$2.$ Press $tan$

$3.$ Press

$1.947368$

$4.$ Press $=$

The result will be:

$62.881890°$

Finally, round off the answer to the nearest minute.

$theta$	$=$	$62.881890°$
	$=$	$62°49’8”$	Press DMS on your calculator
	$=$	$62°49’$	Round down since the seconds is less than $30”$

$62°49’$

Question 3 of 6

A tree

$26 m$ tall casts a shadow

$31 m$ long. What angle

$(theta)$ do the rays of the sun make with the ground? (nearest degree)

$theta=$ (40) $°$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the missing angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{26}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{31}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{26}}{\color{#00880a}{31}}$

$tan theta$	$=$	$0.8387097$
$theta$	$=$	$tan^(-1) 0.8387097$	Get the inverse of $tan$

Simplify this further by evaluating

$tan^(-1) 0.8387097$ using the calculator:

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

$2.$ Press $tan$

$3.$ Press

$0.8387097$

$4.$ Press $=$

The result will be:

$39.986887°$

Finally, round off the answer to the nearest degree.

$theta$	$=$	$39.986887°$
	$=$	$39°59’$	Press DMS on your calculator
	$=$	$40°$	Round up since the minutes is more than $30’$

$40°$

Question 4 of 6

The ray of light from a lighthouse to a boat at sea is

$37.2 m$ long. If the boat is

$29 m$ away from the base of the lighthouse, what angle

$(theta)$ does the ray of light make to the sea level? (nearest degree)

$theta=$ (39) $°$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the missing angle.

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{29}$

$\color{#e85e00}{\text{hypotenuse}}=\color{#e85e00}{37.2}$

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

$cos$ ratio to find

$theta$ .

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

$cos theta$	$=$	$\frac{\color{#00880a}{29}}{\color{#e85e00}{37.2}}$

$theta$	$=$	$cos^(-1) (29/37.2)$	Get the inverse of $cos$

Simplify this further by evaluating

$cos^(-1) (29/37.2)$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$29$

$4.$ Press $÷$

$5.$ Press

$37.2$

$6.$ Press $=$

The result will be:

$38.7788°$

Finally, round off the answer to the nearest degree.

$theta$	$=$	$38.7788°$
	$=$	$38°46’43”$	Press DMS on your calculator
	$=$	$39°$	Round up since the minutes is more than $30’$

$39°$

Question 5 of 6

A ladder is placed

$3 m$ up a wall. The distance between the foot of the ladder and the wall is

$1 m$ . What is the angle of inclination

$(theta)$ of the ladder with the horizontal floor? (nearest degree)

$theta=$ (72) $°$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that the scenario creates a triangle. Label it in reference to the missing angle.

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{3}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{1}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{3}}{\color{#00880a}{1}}$

$tan theta$	$=$	$3$
$theta$	$=$	$tan^(-1) 3$	Get the inverse of $tan$

Simplify this further by evaluating

$tan^(-1) 3$ using the calculator:

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

$2.$ Press $tan$

$3.$ Press

$3$

$4.$ Press $=$

The result will be:

$71.56505°$

Finally, round off the answer to the nearest degree.

$theta$	$=$	$71.56505°$
	$=$	$71°33’54”$	Press DMS on your calculator
	$=$	$72°$	Round up since the minutes is more than $30’$

$72°$

Question 6 of 6

Find the size to the nearest minute of one of the base angles of the iscoseles triangle below.

(69) $°$ (51) $'$

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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

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Notice that a right triangle is formed by the isosceles triangle.

Redraw this triangle separately and label the values.

Let

$theta$ be one of the base angles.

Halve the base of the isosceles triangle to get the adjacent side:

$8divide2=4$

$\color{#004ec4}{\text{opposite}}=\color{#004ec4}{10.9}$

$\color{#00880a}{\text{adjacent}}=\color{#00880a}{4}$

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

$tan$ ratio to find

$theta$ .

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

$tan theta$	$=$	$\frac{\color{#004ec4}{10.9}}{\color{#00880a}{4}}$

$tan theta$	$=$	$2.725$
$theta$	$=$	$tan^(-1) 2.725$	Get the inverse of $tan$

Simplify this further by evaluating

$tan^(-1) 2.725$ using the calculator:

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

$2.$ Press $tan$

$3.$ Press

$2.725$

$4.$ Press $=$

The result will be:

$69.84825°$

Finally, round off the answer to the nearest minute.

$theta$	$=$	$69.84825°$
	$=$	$69°50’58”$	Press DMS on your calculator
	$=$	$69°51’$	Round up since the seconds is more than $30”$

$69°51’$