a The Law of Sines relates the sine of each angle to the length of the opposite side.
B
b Determine the trigonometric ratio to be used according to the given information.
C
c Determine the trigonometric ratio to use according to the given information and the unknown.
A
a x ≈ 10.3
B
b No solution possible because the hypotenuse is the longest side of a right triangle.
C
c x≈5.39
Practice makes perfect
a For any △ ABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.
The Law of Sines relates the sine of each angle to the length of the opposite side.
sin A/a=sin B/b=sin C/c
Let's use this law to find the value of x. Consider the given diagram.
As we can see, we are not given any known angle and opposite side length pairs. However, since we know two of the three interior angles of the triangle, we can find the third interior angle using the Triangle Angle Sum Theorem.
180 - 80^(∘) - 27^(∘) = 73^(∘)
Let's add this angle to our diagram.
We know that the length of one side is 10 and that the measure of its opposite angle is 73^(∘). We want to find the length of the side that is opposite to the angle whose measure is 80^(∘). With the new information we can use the Law of Sines!
sin( 73^(∘))/10 = sin( 80^(∘) )/x
Let's solve the equation for x.
b We are given the length of two legs of a right triangle, and want to find the measure of one of its acute angles.
Notice that we are given the lengths of one of the legs and the length of the hypotenuse. However, based on the diagram, this leg is longer than the hypotenuse. No right triangle has a hypotenuse that is shorter than one of the legs, so there is no value of x that completes the triangle.
c Consider the given diagram.
In order to find x, it would be useful find the length of the other dashed segment from the diagram. Let's denote this distance as y. Then, by the Segment Addition Postulate, y+5 becomes the length of the other leg in the larger right triangle.
Now, notice that the given side is opposite the given angle, and that the side with length y+5 is adjacent to the given angle. Therefore, we can use the tangent ratio to help us find the value of y.
tan (θ) = opposite/adjacent
In our triangle, we have that θ = 30^(∘) and the length of the opposite leg is 6. Let's find the length y by substituting the values in the formula.
