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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}

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