Answer:
Hypotenuse = 159 ft (nearest integer)
[tex]\sin(x)=\dfrac{53}{159}[/tex]
[tex]\cos(x)=\dfrac{150}{159}[/tex]
[tex]\tan(x)=\dfrac{53}{150}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
From inspection of the given right triangle:
Substitute these values into Pythagoras Theorem and solve for c (hypotenuse):
[tex]\implies 53^2+150^2=c^2[/tex]
[tex]\implies 2809+22500=c^2[/tex]
[tex]\implies 25309=c^2[/tex]
[tex]\implies c=\sqrt{25309}[/tex]
[tex]\implies c=159.0880259...[/tex]
[tex]\implies c=159\;\; \sf ft\;\;(nearest\;integer)[/tex]
Therefore, the hypotenuse of the given triangle is 159 ft (nearest integer).
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Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
- θ is the angle.
- O is the side opposite the angle.
- A is the side adjacent the angle.
- H is the hypotenuse (the side opposite the right angle).
Given:
- θ = x
- A = 150
- O = 53
- H = 159
Substitute the values into the ratios:
[tex]\implies \sin(x)=\dfrac{53}{159}[/tex]
[tex]\implies \cos(x)=\dfrac{150}{159}[/tex]
[tex]\implies \tan(x)=\dfrac{53}{150}[/tex]
SOHCAHTOA is a mnemonic for the definitions of the trigonometric ratios applicable to right triangles:
- Sine of an angle is equal to Opposite over Hypotenuse.
- Cosine of an angle is equal to Adjacent over Hypotenuse.
- Tangent of an angle is equal to Opposite over Adjacent.
(See attachment).
