Question

right triangles practice
trigonometry
directions: mark the given information. determine which trig function to use and write a proportion.
solve the proportion.
1.
2.
3.
4.
5.
6.

Explanation:

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Problem 1:

Step1: Identify cosine for hypotenuse

Adjacent to $\angle M$: $63.4$ cm, hypotenuse $z$. Use $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(25^\circ)=\frac{63.4}{z}$

Step2: Rearrange to solve for z

$z=\frac{63.4}{\cos(25^\circ)}$
Calculate $\cos(25^\circ)\approx0.9063$, so $z\approx\frac{63.4}{0.9063}\approx70.0$ cm

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Problem 2:

Step1: Identify cosine for angle y

Adjacent to $\angle y$: $24.85$ cm, hypotenuse $83$ cm. Use $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(y^\circ)=\frac{24.85}{83}=0.2994$

Step2: Solve for y using arccosine

$y=\arccos(0.2994)\approx72.6^\circ$

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Problem 3:

Step1: Identify sine for side y

$\angle D=51^\circ$, hypotenuse $83$ cm, opposite side $y$. Use $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
$\sin(51^\circ)=\frac{y}{83}$

Step2: Rearrange to solve for y

$y=83\times\sin(51^\circ)$
Calculate $\sin(51^\circ)\approx0.7771$, so $y\approx83\times0.7771\approx64.5$ cm

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Problem 4:

Step1: Identify cosine for side x

$\angle B=48^\circ$, hypotenuse $38$ m, adjacent side $x$. Use $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(48^\circ)=\frac{x}{38}$

Step2: Rearrange to solve for x

$x=38\times\cos(48^\circ)$
Calculate $\cos(48^\circ)\approx0.6691$, so $x\approx38\times0.6691\approx25.4$ m

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Problem 5:

Step1: Identify sine for angle x

Opposite to $\angle x$: $100$ cm, hypotenuse $172$ cm. Use $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
$\sin(x^\circ)=\frac{100}{172}\approx0.5814$

Step2: Solve for x using arcsine

$x=\arcsin(0.5814)\approx35.5^\circ$

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Problem 6:

Step1: Identify sine for angle x

Opposite to $\angle x$: $9$ cm, hypotenuse $25$ cm. Use $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
$\sin(x^\circ)=\frac{9}{25}=0.36$

Step2: Solve for x using arcsine

$x=\arcsin(0.36)\approx21.1^\circ$

Answer:

1. $z\approx70.0$ cm
2. $y\approx72.6^\circ$
3. $y\approx64.5$ cm
4. $x\approx25.4$ m
5. $x\approx35.5^\circ$
6. $x\approx21.1^\circ$