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ExtremeBDS [4]
2 years ago
6

Triangle ΔABC has side lengths of a = 18,

qrt{3}" alt="b=18\sqrt{3}" align="absmiddle" class="latex-formula"> and c = 36 inches.
Part A: Determine the measure of angle A

Part B: Show how to use the unit circle to find tan A

Part C: Calculate the area of ΔABC.
Mathematics
1 answer:
ozzi2 years ago
7 0

The measure of angle A is 30 degree, the value of tan A is 1/√3 and the area of triangle ABC is 280.6 squared inches.

<h3 /><h3>What is the law of cosine?</h3>

When the three sides of a triangle is known, then to find any angle, the law of cosine is used.

It can be given as,

\angle A=\cos^{-1}\left(\dfrac{b^2+c^2-a^2}{2bc}\right) \\\angle B=\cos^{-1}\left(\dfrac{a^2+c^2-b^2}{2ac}\right) \\\angle C=\cos^{-1}\left(\dfrac{a^2+b^2-c^2}{2ab}\right)

Here, a,b and c are the sides of the triangle and A,B and C are the angles of the triangle.

Triangle ΔABC has side lengths of a = 18, b=18√3  and c = 36 inches.

  • Part A: Determine the measure of angle A

Put the value, in the cosine law, the measure of angle A.

\angle A=\cos^{-1}\left(\dfrac{b^2+c^2-a^2}{2bc}\right) \\\angle A=\cos^{-1}\left(\dfrac{(18\sqrt{3})^2+36^2-18^2}{2(18\sqrt{3})(36)}\right) \\\angle A=0.5236\rm\; rad\\\angle A=30^o\rm\; degree\\

  • Part B: Show how to use the unit circle to find tan A

Using the chart of unit circle, the value of tangent A can be found out. The tangent A is,

\tan A=\tan 30^o\\\tan A=\dfrac{1}{\sqrt{3}}

  • Part C: Calculate the area of ΔABC.

Use the following formula to find area of ΔABC.,

A=\dfrac{ab.\sin C}{2}\\A=\dfrac{18\times18\sqrt{3}.\sin C}{2}\\A=280.6\rm\; in^2

Thus, the measure of angle A is 30 degree, the value of tan A is 1/√3 and the area of triangle ABC is 280.6 squared inches.

Learn more about the law of cosine here;

brainly.com/question/4372174

#SPJ1

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For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limn→[
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The following are the solution to the given points:

Step-by-step explanation:

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1) \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\2) \sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}

Solve point 1 that is \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\:

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k= 1 \to  s_1 = \frac{1}{1+1} - \frac{1}{1+2}\\\\

                  = \frac{1}{2} - \frac{1}{3}\\\\

k= 2 \to  s_2 = \frac{1}{2+1} - \frac{1}{2+2}\\\\

                  = \frac{1}{3} - \frac{1}{4}\\\\

k= 3 \to  s_3 = \frac{1}{3+1} - \frac{1}{3+2}\\\\

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S=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....\frac{1}{n+1}-\frac{1}{n+2}\\\\

   =\frac{1}{2}-\frac{1}{5}+\frac{1}{n+1}-\frac{1}{n+2}\\\\

When s_n \ \ dt_{n \to 0}

=\frac{1}{2}-\frac{1}{5}+\frac{1}{0+1}-\frac{1}{0+2}\\\\=\frac{1}{2}-\frac{1}{5}+\frac{1}{1}-\frac{1}{2}\\\\= 1 -\frac{1}{5}\\\\= \frac{5-1}{5}\\\\= \frac{4}{5}\\\\

\boxed{\text{In point 1:} \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2} =\frac{4}{5}}

In point 2: \sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}

when,

k= 1 \to  s_1 = \frac{1}{(1+6)(1+7)}\\\\

                  = \frac{1}{7 \times 8}\\\\= \frac{1}{56}

k= 2 \to  s_1 = \frac{1}{(2+6)(2+7)}\\\\

                  = \frac{1}{8 \times 9}\\\\= \frac{1}{72}

k= 3 \to  s_1 = \frac{1}{(3+6)(3+7)}\\\\

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k= n^  \to  s_n = \frac{1}{(n+6)(n+7)}\\\\

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S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(n+6)(n+7)}\\\\

when s_n \ \ dt_{n \to 0}

S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(0+6)(0+7)}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{6 \times 7}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}+\frac{1}{42}\\\\=\frac{45+35+28+60}{2520}\\\\=\frac{168}{2520}\\\\=0.066

\boxed{\text{In point 2:} \sum ^{\infty}_{k = 1} \frac{1}{(n+6)(n+7)} = 0.066}

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