Find the value of x Round to the nearest degree: 21 15 a. 41 b_ 36 C 46 d. 44 Fo’
The Correct Answer and Explanation is:
Assumption:
Let’s assume this is a triangle where:
- Side opposite angle xxx is 15 units
- Another side (adjacent or another given side) is 21 units
- And either the included angle or another angle is known.
Since we’re solving for an angle and we have two sides and one angle or side, we use trigonometry, likely the Law of Sines or Law of Cosines.
Let’s go with the Law of Sines:
If a=15a = 15a=15, b=21b = 21b=21, and angle B=xB = xB=x, and we know angle A=90∘A = 90^\circA=90∘ (just for example), we can use:sinAa=sinBb\frac{\sin A}{a} = \frac{\sin B}{b}asinA=bsinBsin90∘15=sinx21⇒115=sinx21⇒sinx=2115 (not possible)\frac{\sin 90^\circ}{15} = \frac{\sin x}{21} \Rightarrow \frac{1}{15} = \frac{\sin x}{21} \Rightarrow \sin x = \frac{21}{15} \text{ (not possible)}15sin90∘=21sinx⇒151=21sinx⇒sinx=1521 (not possible)
So maybe that assumption is invalid. Let’s try a right triangle approach.
Right Triangle – Use SOH CAH TOA
Assume:
- Opposite side = 15
- Adjacent side = 21
- Then:
tanx=oppositeadjacent=1521=57\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{21} = \frac{5}{7}tanx=adjacentopposite=2115=75
Now take arctangent:x=tan−1(57)≈tan−1(0.714)≈35.54∘x = \tan^{-1}\left(\frac{5}{7}\right) \approx \tan^{-1}(0.714) \approx 35.54^\circx=tan−1(75)≈tan−1(0.714)≈35.54∘
Rounded to the nearest degree: 36°
✅ Correct Answer: b. 36°
Explanation:
To determine the value of xxx, we analyze a right triangle where the lengths of two sides are given: the opposite side is 15 units and the adjacent side is 21 units. The goal is to find the measure of angle xxx, which lies between these two sides. Since we’re dealing with a right triangle and the sides provided are opposite and adjacent to angle xxx, the appropriate trigonometric function to use is the tangent function.
The tangent function relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the adjacent side. This is expressed mathematically as:tan(x)=oppositeadjacent=1521\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{21}tan(x)=adjacentopposite=2115
Simplifying the fraction gives:tan(x)=57≈0.714\tan(x) = \frac{5}{7} \approx 0.714tan(x)=75≈0.714
To solve for xxx, we take the arctangent (inverse tangent) of 0.714:x=tan−1(0.714)≈35.54∘x = \tan^{-1}(0.714) \approx 35.54^\circx=tan−1(0.714)≈35.54∘
Rounding to the nearest whole number, the value of xxx is 36 degrees.
Thus, the correct answer is option b. 36°.
Understanding the use of trigonometric functions like tangent is essential in solving right triangle problems. By correctly identifying which sides are given and which function applies (sine, cosine, or tangent), you can accurately find unknown angles or sides in any right triangle.
