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anyanavicka [17]
2 years ago
12

This is finding exact values of sin theta/2 and tan theta/2. I’m really confused and now don’t have a clue on how to do this, pl

ease help

Mathematics
1 answer:
Lostsunrise [7]2 years ago
3 0

First,

tan(<em>θ</em>) = sin(<em>θ</em>) / cos(<em>θ</em>)

and given that 90° < <em>θ </em>< 180°, meaning <em>θ</em> lies in the second quadrant, we know that cos(<em>θ</em>) < 0. (We also then know the sign of sin(<em>θ</em>), but that won't be important.)

Dividing each part of the inequality by 2 tells us that 45° < <em>θ</em>/2 < 90°, so the half-angle falls in the first quadrant, which means both cos(<em>θ</em>/2) > 0 and sin(<em>θ</em>/2) > 0.

Now recall the half-angle identities,

cos²(<em>θ</em>/2) = (1 + cos(<em>θ</em>)) / 2

sin²(<em>θ</em>/2) = (1 - cos(<em>θ</em>)) / 2

and taking the positive square roots, we have

cos(<em>θ</em>/2) = √[(1 + cos(<em>θ</em>)) / 2]

sin(<em>θ</em>/2) = √[(1 - cos(<em>θ</em>)) / 2]

Then

tan(<em>θ</em>/2) = sin(<em>θ</em>/2) / cos(<em>θ</em>/2) = √[(1 - cos(<em>θ</em>)) / (1 + cos(<em>θ</em>))]

Notice how we don't need sin(<em>θ</em>) ?

Now, recall the Pythagorean identity:

cos²(<em>θ</em>) + sin²(<em>θ</em>) = 1

Dividing both sides by cos²(<em>θ</em>) gives

1 + tan²(<em>θ</em>) = 1/cos²(<em>θ</em>)

We know cos(<em>θ</em>) is negative, so solve for cos²(<em>θ</em>) and take the negative square root.

cos²(<em>θ</em>) = 1/(1 + tan²(<em>θ</em>))

cos(<em>θ</em>) = - 1/√[1 + tan²(<em>θ</em>)]

Plug in tan(<em>θ</em>) = - 12/5 and solve for cos(<em>θ</em>) :

cos(<em>θ</em>) = - 1/√[1 + (-12/5)²] = - 5/13

Finally, solve for sin(<em>θ</em>/2) and tan(<em>θ</em>/2) :

sin(<em>θ</em>/2) = √[(1 - (- 5/13)) / 2] = 3/√(13)

tan(<em>θ</em>/2) = √[(1 - (- 5/13)) / (1 + (- 5/13))] = 3/2

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2 points
saul85 [17]

Given:

Width of rectangular prism = 20 ft

height of rectangular prism = 12 ft

Length of rectangular prism = 45 ft

Cylindrical diameter = 20 ft

To find:

Volume of the tent

Solution:

Volume of the rectangular prism = length × width × height

                                                       = 45 × 20 × 12

Volume of the rectangular prism = 10800 ft³

Radius of cylinder = 20 ÷ 2 = 10 ft

Height of cylinder = length of rectangular prism

Volume of cylinder = \pi r^{2} h

Volume of half cylinder = \frac{1}{2} \pi r^2h

                                      $=\frac{1}{2} \times 3.142\times 10^2 \times45

Volume of half cylinder = 7069.5 ft³

Volume of tent = volume of rectangular prism + volume of half cylinder

                         = 10800 ft³ + 7069.5 ft³

                         = 17869.5 ft³

                         = 17869 ft³ (nearest cubic foot)

The volume of the tent is 17869 cubic feet.

6 0
2 years ago
Please help. How do you find cosine, sine, cosecant and secant with this triangle? ​
Veseljchak [2.6K]

Hi there! You have to remember these 6 basic Trigonometric Ratios which are:

  • sine (sin) = opposite/hypotenuse
  • cosine (cos) = adjacent/hypotenuse
  • tangent (tan) = opposite/adjacent
  • cosecant (cosec/csc) = hypotenuse/opposite
  • secant (sec) = hypotenuse/adjacent
  • cotangent (cot) = adjacent/opposite
  • cosecant is the reciprocal of sine
  • secant is the reciprocal of cosine
  • cotangent is the reciprocal of tangent

Back to the question. Assuming that the question asks you to find the cosine, sine, cosecant and secant of angle theta.

What we have now are:

  • Trigonometric Ratio
  • Adjacent = 12
  • Opposite = 10

Looks like we are missing the hypotenuse. Do you remember the Pythagorean Theorem? Recall it!

  • a²+b² = c²

Define that c-term is the hypotenuse. a-term and b-term can be defined as adjacent or opposite

Since we know the value of adjacent and opposite, we can use the formula to find the hypotenuse.

  • 10²+12² = c²
  • 100+144 = c²
  • 244 = c²

Thus, the hypotenuse is:

\large \boxed{c = 2 \sqrt{61} }

Now that we know all lengths of the triangle, we can find the ratio. Recall Trigonometric Ratio above! Therefore, the answers are:

  • cosine (cosθ) = adjacent/hypotenuse = 12/(2√61) = 6/√61 = <u>(6√61) / 61</u>
  • sine (sinθ) = opposite/hypotenuse = 10/(2√61) = 5/√61 = <u>(5√61) / 61</u>
  • cosecant (cscθ) is reciprocal of sine (sinθ). Hence, cscθ = (2√61/10) = <u>√61/5</u>
  • secant (secθ) is reciprocal of cosine (cosθ). Hence, secθ = (2√61)/12 = <u>√</u><u>61</u><u>/</u><u>6</u>

Questions can be asked through comment.

Furthermore, we can use Trigonometric Identity to find the hypotenuse instead of Pythagorean Theorem.

Hope this helps, and Happy Learning! :)

5 0
3 years ago
You downloaded a video game to your computer. You have a 60-minute free trial of the game. It takes 5 1/6 minutes to set up the
igomit [66]

Answer:

B.

Step-by-step explanation:

Given,

The time for set up the game =  5\frac{1}{6} minutes,  

Also, the time for each level of game is 7\frac{1}{3\\} minutes,  

If there are l levels of games,  

Then the time taken in all levels of game = (7\frac{1}{3}) l<em> </em>

Hence, the total time ( in minutes ) = Set up time + time in all level

= 5\frac{1}{6} +(7\frac{1}{3}) l

We have 60-minute of free trial of the game,

So, the total time taken can not be exceed to 60 minutes,

= 5\frac{1}{6} +(7\frac{1}{3}) l\leq 60

Which is the required inequality.  

Option 'B' is correct.

5 0
2 years ago
Find three solutions of the equation y=9x-4?
Bas_tet [7]
<span>answer
A) (-5,-49),(-2,-22),(3,23)

</span>when x = - 5 ; y=9x-4 = 9(-5) -4  = -49
when x = - 2 ; y=9x-4 = 9(-2) -4  = -22
when x = 3 ; y=9x-4 = 9(3) -4  = 23
6 0
3 years ago
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Heather has a coin that has a 60\%60%60, percent chance of showing heads each time it is flipped. She is going to flip the coin
Elza [17]

Answer:

(5/3) (0.60)^3(0.40)^2

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
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