Math

QuestionFind secθ\sec \theta if sinθ=49\sin \theta = -\frac{4}{9} and tanθ>0\tan \theta > 0.

Studdy Solution

STEP 1

Assumptions1. The given value of sinθ\sin \theta is 49-\frac{4}{9} . The given condition is tanθ>0\tan \theta >0
3. We need to find the value of secθ\sec \theta

STEP 2

First, we need to find the value of cosθ\cos \theta. We use the Pythagorean identity for sine and cosine, which is sin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta =1.
Rearranging for cosθ\cos \theta, we getcosθ=±1sin2θ\cos \theta = \pm \sqrt{1 - \sin^2 \theta}

STEP 3

Plug in the given value for sinθ\sin \theta to calculate cosθ\cos \theta.
cosθ=±1(9)2\cos \theta = \pm \sqrt{1 - \left(-\frac{}{9}\right)^2}

STEP 4

implify the expression inside the square root.
cosθ=±1(1681)\cos \theta = \pm \sqrt{1 - \left(\frac{16}{81}\right)}

STEP 5

Subtract the fractions under the square root.
cosθ=±81811681\cos \theta = \pm \sqrt{\frac{81}{81} - \frac{16}{81}}

STEP 6

implify the fraction under the square root.
cosθ=±6581\cos \theta = \pm \sqrt{\frac{65}{81}}

STEP 7

Take the square root of the fraction.
cosθ=±659\cos \theta = \pm \frac{\sqrt{65}}{9}

STEP 8

We know that tanθ>0\tan \theta >0. In the unit circle, tangent is positive in the first and third quadrants. However, since sinθ<0\sin \theta <0, we know that θ\theta must be in the third quadrant where cosine is negative. So, we choose the negative root for cosθ\cos \theta.
cosθ=65\cos \theta = -\frac{\sqrt{65}}{}

STEP 9

Now that we have the value of cosθ\cos \theta, we can find the value of secθ\sec \theta. The secant function is the reciprocal of the cosine function, sosecθ=cosθ\sec \theta = \frac{}{\cos \theta}

STEP 10

Plug in the value for cosθ\cos \theta to calculate secθ\sec \theta.
secθ=659\sec \theta = \frac{}{-\frac{\sqrt{65}}{9}}

STEP 11

implify the fraction to find the exact value of secθ\sec \theta.
secθ=965\sec \theta = -\frac{9}{\sqrt{65}}So, the exact value of secθ\sec \theta is 965-\frac{9}{\sqrt{65}}.

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