1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mario62 [17]
3 years ago
8

Given tan theta =9, use trigonometric identities to find the exact value of each of the following:_______

Mathematics
1 answer:
Ludmilka [50]3 years ago
4 0

Answer:

(a)\ \sec^2(\theta) = 82

(b)\ \cot(\theta) = \frac{1}{9}

(c)\ \cot(\frac{\pi}{2} - \theta) = 9

(d)\ \csc^2(\theta) = \frac{82}{81}

Step-by-step explanation:

Given

\tan(\theta) = 9

Required

Solve (a) to (d)

Using tan formula, we have:

\tan(\theta) = \frac{Opposite}{Adjacent}

This gives:

\frac{Opposite}{Adjacent} = 9

Rewrite as:

\frac{Opposite}{Adjacent} = \frac{9}{1}

Using a unit ratio;

Opposite = 9; Adjacent = 1

Using Pythagoras theorem, we have:

Hypotenuse^2 = Opposite^2 + Adjacent^2

Hypotenuse^2 = 9^2 + 1^2

Hypotenuse^2 = 81 + 1

Hypotenuse^2 = 82

Take square roots of both sides

Hypotenuse =\sqrt{82}

So, we have:

Opposite = 9; Adjacent = 1

Hypotenuse =\sqrt{82}

Solving (a):

\sec^2(\theta)

This is calculated as:

\sec^2(\theta) = (\sec(\theta))^2

\sec^2(\theta) = (\frac{1}{\cos(\theta)})^2

Where:

\cos(\theta) = \frac{Adjacent}{Hypotenuse}

\cos(\theta) = \frac{1}{\sqrt{82}}

So:

\sec^2(\theta) = (\frac{1}{\cos(\theta)})^2

\sec^2(\theta) = (\frac{1}{\frac{1}{\sqrt{82}}})^2

\sec^2(\theta) = (\sqrt{82})^2

\sec^2(\theta) = 82

Solving (b):

\cot(\theta)

This is calculated as:

\cot(\theta) = \frac{1}{\tan(\theta)}

Where:

\tan(\theta) = 9 ---- given

So:

\cot(\theta) = \frac{1}{\tan(\theta)}

\cot(\theta) = \frac{1}{9}

Solving (c):

\cot(\frac{\pi}{2} - \theta)

In trigonometry:

\cot(\frac{\pi}{2} - \theta) = \tan(\theta)

Hence:

\cot(\frac{\pi}{2} - \theta) = 9

Solving (d):

\csc^2(\theta)

This is calculated as:

\csc^2(\theta) = (\csc(\theta))^2

\csc^2(\theta) = (\frac{1}{\sin(\theta)})^2

Where:

\sin(\theta) = \frac{Opposite}{Hypotenuse}

\sin(\theta) = \frac{9}{\sqrt{82}}

So:

\csc^2(\theta) = (\frac{1}{\frac{9}{\sqrt{82}}})^2

\csc^2(\theta) = (\frac{\sqrt{82}}{9})^2

\csc^2(\theta) = \frac{82}{81}

You might be interested in
The volume of a cylinder is 8177 cm³. If the radius is 3 cm, what is the height
Virty [35]

Answer:

None of the options are correct

О E. 289.2 cm

Step-by-step explanation:

Volume of a cylinder = v = л r² h

л = 3.1416     (aprox)

r = radius

h = height

8177 = 3.1416 * 3² * h

8177 = 3.1416 * 9 * h

8177 = 28.2744 * h

8177/282744 = h

h = 289.2 cm

4 0
2 years ago
What does (r-c) represent in this context?
gavmur [86]
The correct answer is C
5 0
2 years ago
Read 2 more answers
I will give brainiest to whoever answers correctly !!
Viktor [21]

Answer: 10k

Step-by-step explanation:

6 0
3 years ago
two rectangular properties share a common side. Lot is 33 feet wide and 42 feet long. The combined area of the lots is 1,848 squ
klio [65]

First, find the area of lot A.

To find the area, multiply the length by the width.

33*42=1,386

Now, subtract the area of lot a from the total area. This will give us the area of lot b.

1,848-1,386=462

The area of lot b is 462 square feet.

Finally, divide the area of lot b (462) by its length to find the width. Since they both share a side, we know that it’s length is 42 feet.

462/42=11

Lot b is 11 feet wide.

Hope this helps!

4 0
3 years ago
Which side lengths below DO NOT produce a right triangle?
tresset_1 [31]
C 3*3 = 9 6*6=36 and 36 plus 9 is 45 and 9*9 = 81 so it’s not equal and not right.
3 0
3 years ago
Read 2 more answers
Other questions:
  • Simplify 8(7x-1)+4(x+5)
    14·1 answer
  • A bride wants to serve her 175 guests cheesecake instead of wedding cake. She purchases 14 cheesecakes that serve 12 people each
    15·2 answers
  • If DE = 37 cm and EF = 16 cm, then what are the possible lengths for DF so that DE,EF, and DF can form a triangle? Explain your
    5·2 answers
  • What is the value of 4P -2 when P= 8​
    10·2 answers
  • A cube has a volume of 1,000 cubic feet what is the length on each side of the cube
    9·1 answer
  • Pls answer quick rjwejriwejriwekr
    14·2 answers
  • No links or i'll report
    15·1 answer
  • I need help guys ASAP
    11·1 answer
  • In a box containing a dozen eggs ,3 eggs are rotten .What fraction of eggs are not rotten?​
    12·1 answer
  • You are about to list your house as For Sale By Owner. Suppose you need to allow
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!