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
GalinKa [24]
3 years ago
6

Cosθ=−2√3 , where π≤θ≤3π2 .

Mathematics
1 answer:
Alex787 [66]3 years ago
8 0

Answer:

sin(\theta + \beta) = -\frac{\sqrt{7}}{5}-4\frac{\sqrt{2}}{15}

Step-by-step explanation:

step 1

Find the  sin(\theta)

we know that

Applying the trigonometric identity

sin^2(\theta)+ cos^2(\theta)=1

we have

cos(\theta)=-\frac{\sqrt{2}}{3}

substitute

sin^2(\theta)+ (-\frac{\sqrt{2}}{3})^2=1

sin^2(\theta)+ \frac{2}{9}=1

sin^2(\theta)=1- \frac{2}{9}

sin^2(\theta)= \frac{7}{9}

sin(\theta)=\pm\frac{\sqrt{7}}{3}

Remember that

π≤θ≤3π/2

so

Angle θ belong to the III Quadrant

That means ----> The sin(θ) is negative

sin(\theta)=-\frac{\sqrt{7}}{3}

step 2

Find the sec(β)

Applying the trigonometric identity

tan^2(\beta)+1= sec^2(\beta)

we have

tan(\beta)=\frac{4}{3}

substitute

(\frac{4}{3})^2+1= sec^2(\beta)

\frac{16}{9}+1= sec^2(\beta)

sec^2(\beta)=\frac{25}{9}

sec(\beta)=\pm\frac{5}{3}

we know

0≤β≤π/2 ----> II Quadrant

so

sec(β), sin(β) and cos(β) are positive

sec(\beta)=\frac{5}{3}

Remember that

sec(\beta)=\frac{1}{cos(\beta)}

therefore

cos(\beta)=\frac{3}{5}

step 3

Find the sin(β)

we know that

tan(\beta)=\frac{sin(\beta)}{cos(\beta)}

we have

tan(\beta)=\frac{4}{3}

cos(\beta)=\frac{3}{5}

substitute

(4/3)=\frac{sin(\beta)}{(3/5)}

therefore

sin(\beta)=\frac{4}{5}

step 4

Find sin(θ+β)

we know that

sin(A + B) = sin A cos B + cos A sin B

so

In this problem

sin(\theta + \beta) = sin(\theta)cos(\beta)+ cos(\theta)sin (\beta)

we have

sin(\theta)=-\frac{\sqrt{7}}{3}

cos(\theta)=-\frac{\sqrt{2}}{3}

sin(\beta)=\frac{4}{5}

cos(\beta)=\frac{3}{5}

substitute the given values in the formula

sin(\theta + \beta) = (-\frac{\sqrt{7}}{3})(\frac{3}{5})+ (-\frac{\sqrt{2}}{3})(\frac{4}{5})

sin(\theta + \beta) = (-3\frac{\sqrt{7}}{15})+ (-4\frac{\sqrt{2}}{15})

sin(\theta + \beta) = -\frac{\sqrt{7}}{5}-4\frac{\sqrt{2}}{15}

You might be interested in
Factor this equation: x^2-8xy^2+y^4​
Fofino [41]
(x + 4y) • (x - 12y)
4 0
3 years ago
from the edge of a cliff 180m high, the angles of depression of two boats at sea are 43° and 52° respectively what is the distan
MrRa [10]

tan 52 = 180/a; a = 180/tan 52

tan 43 = 180/b; b = 180/tan 43

you need to calculate, a-b = 180/tan 52 - 180/tan 43

substitute the value of tan thetas, and calculate it simply...


8 0
3 years ago
I need help with 15,19,17<br><br> Negative exponents
Hoochie [10]
15) 4m^2
17)9n^6
198a^2 (^ = power of)
6 0
2 years ago
Which is the larger amount 4/10 of a dollar or 8/10 of a dollar
3241004551 [841]
8/10 of a dollar is the correct answer

7 0
3 years ago
Read 2 more answers
Which expression is the simplified form of -10 + 3x - x - 7x + 5x + 7 + 2x?
Mars2501 [29]

Answer:

B

Step-by-step explanation:

-10 + 3x - x - 7x + 5x + 7 + 2x

combine like terms

-10 + 3x - x - 7x + 5x + 7 + 2x

-3 + 3x - x - 7x + 5x + 2x

the -7x and the 5x + 2x cancel out to 0, so for the x terms we are left with 3x -x to get 2x

= 2x - 3

8 0
3 years ago
Other questions:
  • Can someoneon help me please I dont understand<br>this.
    6·1 answer
  • (10 points) State the relationship between segments MA and TH. Explain.
    6·1 answer
  • Latoya is on her way home in her car. She has driven 30 miles so far, which is five-sixths of the way home. What is the total le
    11·2 answers
  • Tan45degree - 10cos60degree
    9·1 answer
  • What is the simplified form of x + 3/3 + x + 6/3
    7·2 answers
  • Please help me find the value of X
    8·2 answers
  • Solve for x and then find the measure of ZB
    10·1 answer
  • Choose one answer is math
    10·1 answer
  • What is the range of this set of heights in centimeters?
    13·1 answer
  • A bag contains 42 red, 45 green, 20 yellow, and 32 pirple candies. You pick one camdy at random. Find the probability that it is
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!