Question

In questions 16-23 solve $\mathrm{\Delta }\mathrm{XYZ}$ given $Z={46}^{\circ },\mathrm{YZ}=30\mathrm{m}$ and $\mathrm{XY}=25\mathrm{m}$.

Short answer

Question: Find all the missing angles and sides of triangle XYZ given that angle Z = ${46}^{\circ }$, side YZ = 30 m, and side XY = 25 m. Answer: In triangle XYZ, angle X = ${43.23}^{\circ }$, angle Y = ${90.77}^{\circ }$, side XY = 25 m, side YZ = 30 m, and side XZ = $38.45\text{m}$.

Step-by-step solution

Identify the known values

We have: - Angle Z = ${46}^{\circ }$ - Side YZ (opposite angle Z) = 30 m - Side XY (opposite angle X) = 25 m

Apply the Law of Sines to find angle X

Using the Law of Sines: $$\frac{\sin X}{XY}=\frac{\sin Z}{YZ}$$ Substitute the given values: $$\frac{\sin X}{25}=\frac{\sin {46}^{\circ }}{30}$$ Now, isolate sin X: $$\sin X=\frac{\sin {46}^{\circ }}{30}\times 25$$ Find the value of sin X: $$\sin X\approx 0.684$$ Now, find X by taking the inverse sine of 0.684: $$X\approx {43.23}^{\circ }$$

Find angle Y using the triangle angle sum property

The sum of the angles in a triangle is always equal to ${180}^{\circ }$. Therefore, we can find angle Y by subtracting angle X and angle Z from ${180}^{\circ }$: $$Y={180}^{\circ }-X-Z$$ Substitute the values of angle X and Z: $$Y={180}^{\circ }-{43.23}^{\circ }-{46}^{\circ }$$ Calculate the value of angle Y: $$Y\approx {90.77}^{\circ }$$

Apply the Law of Sines to find side XZ

Using the Law of Sines again: $$\frac{\sin Y}{XZ}=\frac{\sin Z}{YZ}$$ Substitute the given values and the calculated value of angle Y: $$\frac{\sin {90.77}^{\circ }}{XZ}=\frac{\sin {46}^{\circ }}{30}$$ Now, isolate XZ: $$XZ=\frac{30\times \sin {90.77}^{\circ }}{\sin {46}^{\circ }}$$ Find the value of side XZ: $$XZ\approx 38.45\text{m}$$ Now, we have found all the missing angles and sides. The triangle XYZ has angles X = ${43.23}^{\circ }$, Y = ${90.77}^{\circ }$, Z = ${46}^{\circ }$, and sides XY = 25 m, YZ = 30 m, and XZ = $38.45\text{m}$.

Key concepts

Law of Sines

When solving triangles, particularly ones that are not right-angled, the Law of Sines is an invaluable tool. It expresses a relationship between the lengths of the sides of a triangle and the sines of its angles. According to the Law of Sines,
$$\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}$$where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the sides opposite these angles respectively.
In our exercise, we used this principle to calculate one of the unknown angles, angle X. After identifying the known side lengths and angle, we set up a proportion using the law that involved the sides opposite the known and unknown angles. Solving for the sine of angle X, we gained a value that would help us find the actual measure of the angle with the aid of inverse trigonometric functions.

It's critical to ensure that the angle-side pairs are opposite to each other in this law, as mixing them up can lead to incorrect results. Furthermore, this law comes in very handy in the case of 'SSA' or 'AAS' triangle problems, where two sides and a non-included angle or two angles and a non-included side are given.

Triangle Angle Sum Property

The triangle angle sum property is a fundamental concept in geometry. It states that the sum of the angles inside any triangle always adds up to $$\text{Double exponent: use braces to clarify}$$.This property held the key to finding the third angle in our problem, angle Y. Once we calculated angle X using the Law of Sines, we determined angle Y by subtracting the measures of angles X and Z from $$\text{Double exponent: use braces to clarify}$$ thus revealing the measures of all three angles. This property is useful for all triangles, regardless of their type—acute, obtuse, or right-angled—and is often used in concert with other properties or theorems, like the Law of Sines, to solve for unknown components of the triangle.

Inverse Sine

Inverse sine, also known as sinusoidal or arcsine, is a function that reverses the operation of the sine function. It takes a ratio (typically of a side length to the hypotenuse in a right-angled triangle) and returns an angle measure as the output. In our exercise, after finding the sine of angle X, we sought the actual angle. This is where the inverse sine comes into the equation. Symbolized as $${\sin }^{-1}$$, it's used to determine the angle whose sine is the ratio we have calculated.
$$\text{Double exponent: use braces to clarify}$$
This function is essential for solving triangles when a side and an angle opposite to it are known, but the angle itself is unknown. However, care must be taken with the range of values for which the inverse sine is defined, as it returns values in the range from $$\text{Double exponent: use braces to clarify}$$ to $$\text{Double exponent: use braces to clarify}$$ A calculator or computer algebra system that correctly handles this function is vital for accurately solving exercises like the one in our example.