Answer from 1.1.1 to 1.1.6😭

Question

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Okay, Ntombenhle.💋 Let's tackle all these questions from the diagram.

1.1.1 Write down the length of FG in terms of $a$ and $\phi$ .

Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the adjacent side (FG), and the angle ( $\phi$ ).
The tangent function relates the opposite and adjacent sides.
$\tan(\phi) = \frac{\text{opposite}}{\text{adjacent}} = \frac{EF}{FG}$
Substitute the given values and solve for FG.
$\tan(\phi) = \frac{a}{FG}$ $FG = \frac{a}{\tan(\phi)}$
The length of FG is $\boxed{\mathbf{\frac{a}{\tan(\phi)}}}$

1.1.2 Write down the length of EG in terms of $a$ and $\phi$ .
Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the hypotenuse (EG), and the angle ( $\phi$ ).
The sine function relates the opposite side and the hypotenuse.
$\sin(\phi) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EF}{EG}$
Substitute the given values and solve for EG.
$\sin(\phi) = \frac{a}{EG}$ $EG = \frac{a}{\sin(\phi)}$
The length of EG is $\boxed{\mathbf{\frac{a}{\sin(\phi)}}}$

1.1.3 If $a = 12$ and $\phi = 35^\circ$ , calculate the length of FG.
Use the formula for FG derived in 1.1.1.
$FG = \frac{a}{\tan(\phi)}$
Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.
$FG = \frac{12}{\tan(35^\circ)}$ $FG \approx \frac{12}{0.7002075}$ $FG \approx 17.138$
The length of FG is $\boxed{\mathbf{17.14}}$ (rounded to two decimal places).

1.1.4 If $a = 12$ and $\phi = 35^\circ$ , calculate the length of EG.
Use the formula for EG derived in 1.1.2.
$EG = \frac{a}{\sin(\phi)}$
Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.
$EG = \frac{12}{\sin(35^\circ)}$ $EG \approx \frac{12}{0.5735764}$ $EG \approx 20.920$
The length of EG is $\boxed{\mathbf{20.92}}$ (rounded to two decimal places).

1.1.5 If $FG = 20$ and $\phi = 40^\circ$ , calculate the length of EF.
Use the tangent ratio
$\tan(\phi) = \frac{EF}{FG}$ .
Substitute $FG = 20$ and $\phi = 40^\circ$ and solve for EF.
$\tan(40^\circ) = \frac{EF}{20}$ $EF = 20 \times \tan(40^\circ)$ $EF \approx 20 \times 0.8390996$ $EF \approx 16.781$
The length of EF is $\boxed{\mathbf{16.78}}$ (rounded to two decimal places).

1.1.6 If $EG = 30$ and $\phi = 50^\circ$ , calculate the length of EF.
Use the sine ratio
$\sin(\phi) = \frac{EF}{EG}$ .
Substitute $EG = 30$ and $\phi = 50^\circ$ and solve for EF.
$\sin(50^\circ) = \frac{EF}{30}$ $EF = 30 \times \sin(50^\circ)$ $EF \approx 30 \times 0.7660444$ $EF \approx 22.981$
The length of EF is $\boxed{\mathbf{22.98}}$ (rounded to two decimal places).

Answer from 1.1.1 to 1.1.6😭

Step-by-step solution

Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the adjacent side (FG), and the angle ( $\phi$ ).

Substitute the given values and solve for FG.

Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the hypotenuse (EG), and the angle ( $\phi$ ).

Substitute the given values and solve for EG.

Use the formula for FG derived in 1.1.1.

Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.

Use the formula for EG derived in 1.1.2.

Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.

Use the tangent ratio

Substitute $FG = 20$ and $\phi = 40^\circ$ and solve for EF.

Use the sine ratio

Substitute $EG = 30$ and $\phi = 50^\circ$ and solve for EF.

Still stuck on this one?

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Answer from 1.1.1 to 1.1.6😭

Step-by-step solution

Identify the trigonometric ratio relating the opposite side (EF = aaa), the adjacent side (FG), and the angle (ϕ\phiϕ).

Substitute the given values and solve for FG.

Identify the trigonometric ratio relating the opposite side (EF = aaa), the hypotenuse (EG), and the angle (ϕ\phiϕ).

Substitute the given values and solve for EG.

Use the formula for FG derived in 1.1.1.

Substitute a=12a = 12a=12 and ϕ=35∘\phi = 35^\circϕ=35∘ and calculate.

Use the formula for EG derived in 1.1.2.

Substitute a=12a = 12a=12 and ϕ=35∘\phi = 35^\circϕ=35∘ and calculate.

Use the tangent ratio

Substitute FG=20FG = 20FG=20 and ϕ=40∘\phi = 40^\circϕ=40∘ and solve for EF.

Use the sine ratio

Substitute EG=30EG = 30EG=30 and ϕ=50∘\phi = 50^\circϕ=50∘ and solve for EF.

Still stuck on this one?

More Mathematics Questions

Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the adjacent side (FG), and the angle ( $\phi$ ).

Identify the trigonometric ratio relating the opposite side (EF = $a$ ), the hypotenuse (EG), and the angle ( $\phi$ ).

Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.

Substitute $a = 12$ and $\phi = 35^\circ$ and calculate.

Substitute $FG = 20$ and $\phi = 40^\circ$ and solve for EF.

Substitute $EG = 30$ and $\phi = 50^\circ$ and solve for EF.