Question

Find the measure of angle X in the triangle below.

Answer 1

Answer:

$\displaystyle 89,91969885°, 90,08030115° ≈ x$

Step-by-step explanation:

In this SOMEWHAT right triangle, we need to use the Law of Sines to solve for the m∠X:

Solving for Angle Measures

$\displaystyle \frac{sin∠C}{c} = \frac{sin∠B}{b} = \frac{sin∠A}{a}$

* In the end, you must use the inverse function, or else you will throw off your answer − $\displaystyle arcsin\:[or\:sin^{-1}].$

Solving for Sides

$\displaystyle \frac{c}{sin∠C} = \frac{b}{sin∠B} = \frac{a}{sin∠A}$

Alright. Let us solve this triangle:

$\displaystyle \frac{sin\:67,38°}{24} = \frac{sin∠X}{26} → \frac{[sin\:67,38°][26]}{24} = sin∠X \\ \\ \\ -arcsin\:\frac{[sin\:67,38°][26]}{24} + 180° ≈ 90,08030115° \\ arcsin\:\frac{[sin\:67,38°][26]}{24} ≈ 89,91969885° \\ \\ OR \\ \\ -sin^{-1}\:\frac{[sin\:67,38°][26]}{24} + 180° ≈ 90,08030115° \\ sin^{-1}\:\frac{[sin\:67,38°][26]}{24} ≈ 89,91969885°$

* This is an extra step further because we are dealing with an actual right triangle, therefore the m∠X will have two angle measures.

I am joyous to assist you anytime.

Answer 2

Answer:

  X = 89.92° or 90.08°

Step-by-step explanation:

The law of sines can be used to find the value of angle X:

  sin(X)/26 = sin(67.38°)/24

  sin(X) = (26/24)sin(67.38°) ≈ 0.99999901787

There are two values of X that have this sine:

  X = arcsin(0.99999901787) ≈ 89.92°

  X = 180° -arcsin(0.99999901787) ≈ 90.08°

There are two solutions: X = 89.92° or 90.08°.

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Comment on the problem

We suspect that the angle is supposed to be considered to be 90°. However, the given angle is reported to 2 decimal places, so we figure the requested angle should also be reported to 2 decimal places.

The lengths of the short side that correspond to the above angles are 10.03 and 9.97 units. If the short side were considered to be 10 units, the triangle would be a right triangle, and the larger acute angle would be ...

  arcsin(24/26) ≈ 67.38014° . . . . rounds to 67.38°

This points up the difficulty of trying to use the Law of Sines on a triangle that is actually a right triangle.