6.1.1

Example One: Determining Height to Attach a Guy Wire to a Pole

• Objective: Determine the height at which to attach a guy wire to a pole that is 15 meters long and positioned at an angle of 40 degrees from the ground.

• Sketch:   - Visual representation includes:     - Ground     - Pole standing straight up     - Guy wire providing stability to the pole

• Triangle Configuration:   - The scenario forms a right triangle where:     - The pole represents the vertical side (height H).     - The guy wire (15 meters) is the hypotenuse.     - The angle between the ground and the wire is 40 degrees.

• Definitions of Triangle Sides:   - Hypotenuse: The side opposite the right angle (the guy wire, 15 meters).   - Opposite: The side opposite the angle of interest (the height H, above the ground).   - Adjacent: The side next to the angle of interest (the horizontal distance from the base of the pole).

• Identifying the Trigonometric Ratio:   - The goal is to solve for H using the known hypotenuse (15 meters) and the opposite side (H).   - Referring to SOHCAHTOA (sine, cosine, tangent):     - Since we need the opposite side (H) and we have the hypotenuse (15), we will use the sine function:

$\sin(40^{\circ}) = \frac{H}{15}$

• Solving for H:   - Rearranging the equation:     - Multiply both sides by 15:

$H = 15 \cdot \sin(40^{\circ})$

• Calculated Height:   - Perform the calculation:     - Using a calculator, $H \approx 9.6 \, \text{meters}$
    - Conclusion: The guy wire should be attached approximately 9.6 meters above the ground.

Example Two: Determining the Height of a Cliff

• Objective: Jared determines the height of a cliff after walking 15 meters from the base and noting that the top of the cliff is at a 65-degree angle above the horizontal.

• Sketch:   - Visual elements include:     - Ground     - Cliff     - Jared's line of sight to the top of the cliff forming another right triangle.

• Triangle Configuration:   - Known information includes:     - Jared's distance from the base of the cliff: 15 meters.     - The angle from the horizontal: 65 degrees.

• Definitions of Triangle Sides:   - Hypotenuse: Line of sight from Jared to the top of the cliff (unknown).   - Opposite: The height of the cliff (H).   - Adjacent: The horizontal length (15 meters from the base).

• Identifying the Trigonometric Ratio:   - To find the height H, we will use the tangent ratio since we need opposite (H) and adjacent (15 meters):

$\tan(65^{\circ}) = \frac{H}{15}$

• Solving for H:   - Rearranging the equation:     - Multiply both sides by 15:

$H = 15 \cdot \tan(65^{\circ})$

• Calculated Height:   - Execute the calculation:     - Resulting in $H \approx 32 \, \text{meters}$   - Conclusion: The height of the cliff is 32 meters.

Example Three: Determining the Length of a Ramp

• Objective: Jazmin seeks to construct a ramp for her motorbike, needing to find the length of the board when the ramp will be inclined at 30 degrees and the truck bed is 1.5 meters above the ground.

• Sketch:   - Visual elements include:     - Ground     - Ramp inclined at 30 degrees

• Triangle Configuration:   - Known information:     - Height of the truck bed (height of ramp): 1.5 meters.     - Angle of the ramp with the horizontal: 30 degrees.     - Length of the ramp: unknown (X).

• Definitions of Triangle Sides:   - Hypotenuse: The ramp length (X, unknown).   - Opposite: The height of the truck bed (1.5 meters).   - Adjacent: The base of the ramp (unknown).

• Identifying the Trigonometric Ratio:   - For this calculation, we will use the sine function since it relates the opposite side (1.5 meters) to the hypotenuse (X):

$\sin(30^{\circ}) = \frac{1.5}{X}$

• Solving for X:   - Rearranging requires multiplying both sides by X first:

$X \cdot \sin(30^{\circ}) = 1.5$   - Next, divide both sides by $\sin(30^{\circ})$:

$X = \frac{1.5}{\sin(30^{\circ})}$

• Calculated Length of the Ramp:   - Perform the calculation:     - Resulting in $X \approx 3 \, \text{meters}$   - Conclusion: The required length of the ramp is 3 meters.