Question

Trigonometric ratios

Answer 1

`\displaystyle 41 \approx x`

Explanation:

`\displaystyle (x)/(35) = csc\:59 \hookrightarrow 35csc\:59 = x; 40,832168903... = x \\ \\ \boxed{41 \approx x}`

OR

`\displaystyle (35)/(x) = sin\:59 \hookrightarrow xsin\:59 = 35 \hookrightarrow (35)/(sin\:59) = x; 40,832168903... = x \\ \\ \boxed{41 \approx x}`

Information on trigonometric ratios

`\displaystyle (OPPOCITE)/(HYPOTENUSE) = sin\:θ \\ (ADJACENT)/(HYPOTENUSE) = cos\:θ \\ (OPPOCITE)/(ADJACENT) = tan\:θ \\ (HYPOTENUSE)/(ADJACENT) = sec\:θ \\ (HYPOTENUSE)/(OPPOCITE) = csc\:θ \\ (ADJACENT)/(OPPOCITE) = cot\:θ`

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Answer 2

`\boxed {\boxed {\sf x \approx 41}}}`

Explanation:

Remember the three trigonometric ratios:

• sinθ= opposite/hypotenuse
• cosθ=adjacent/hypotenuse
• tanθ=opposite/adjacent

Examine the triangle. We have an angle measuring 59°.

The side measuring 35 is opposite the angle. x is the hypotenuse because it is the longest side.

Since we have the opposite side and the hypotenuse, we use sine.

`sin \theta= \frac {opposite}{hypotenuse}`

`sin 59=(35)/(x)`

Now, solve for x by isolating the variable. We can cross multiply. Multiply the first numerator by the second denominator. Then, multiply the first denominator by the second numerator.

`\frac {sin59}{1}=(35)/(x)`

`sin59*x=35*1 \\`

`sin59*x=35`

x is being multiplied by sin 59. The inverse of multiplication is division. Divide both sides by sin59.

`\frac {sin59*x}{sin59}=\frac {35}{sin59}`

`x= \frac {35}{0.8571673007} \\`

`x=40.8321689`

Let's round to the nearest whole number.

• 40.8321689

The 8 in the tenths place tells us to the 0 to a 1.

`x \approx 41`

The hypotenuse of the triangle is approximately 41.