Question

PRECALCULUS PLEASE HELP

In the triangle, the length of side b is 7 the m∠A=60°. Find the exact lengths of a and c. Exact means you CANNOT give me decimals. You will need to give me either an integer a fraction or a radical. Use the unit circle. There will need to be two separate equations that you solve. Show all your work.

Answer 1

Let a = x and c = y

Using the Law of Cosines:

x^2 = 7^2 + y^2 - 2(7)(y)cos60°

x^2 = 49 + y^2 - 14y

x^2 - y^2 + 14y = 49

(x - 7y)(x + 7y) = 49

x - 7y = ±7

x + 7y = ±7

Adding the two equations:

2x = ±14

x = ±7

Substituting x = ±7 into x - 7y = ±7:

±7 - 7y = ±7

-7y = 0

y = 0

Therefore, a = ±7 and c = 0

Answer 2

Using the 30-60-90 triangle ratio and the Pythagorean theorem, we find that side a is 7√3/3 and side c is 14√3/3 in a triangle with side b as 7 and angle A as 60°.

Step-by-step explanation:

To find the lengths of sides a and c in a triangle where side b is 7 and m∠A is 60°, we can use the Pythagorean theorem and trigonometric identities associated with a 60° angle in a right triangle.

Firstly, since the angle A is 60°, this suggests that we may be dealing with a 30-60-90 special right triangle, where the ratios between the lengths of the sides are fixed. Specifically, if the shorter leg (opposite the 30° angle) has length x, the hypotenuse will have length 2x, and the longer leg (opposite the 60° angle) has length x√3.

In this scenario, side b, opposite the 60° angle, corresponds to the longer leg, so b = x√3. We can set up an equation to solve for x:

7 = x√3

x = 7/√3

x = 7√3/3 (rationalized denominator)

Now, side a, which is opposite the 30° angle (therefore it is the shorter leg), will be equal to x and side c, the hypotenuse, will be equal to 2x.

a = x = 7√3/3

c = 2x = 2(7√3/3) = 14√3/3