Question

Suppose sin theta=x/9 and the angle theta is in the 1st quadrant. Write algebraic expressions for cos(theta) and tan(theta) in terms of x.

cos(theta)=

tan(theta)=

Answer 1

We'll consider that x > 0. By the Pythagorean trigonometric identity, we have:


`\sin^2\theta+\cos^2\theta=1\\\\ \left((x)/(9)\right)^2+\cos^2\theta=1\\\\ (x^2)/(81)+\cos^2\theta=1\\\\ \cos^2\theta=1-(x^2)/(81)\\\\ \cos^2\theta=(81-x^2)/(81)\\\\ \cos\theta=\pm\sqrt{(81-x^2)/(81)}\\\\\cos\theta=\pm(√(81-x^2))/(9)`

Since θ is in the 1st quadrant,
`\cos\theta\ \textgreater \ 0`. So:


`\boxed{\cos\theta=(√(81-x^2))/(9)}`

Now, we'll find the tangent. Using the formula below:


`\tan\theta=(\sin\theta)/(\cos\theta)\\\\ \tan\theta=((x)/(9))/(~~(√(81-x^2))/(9)~~)=(x)/(9)*(9)/(√(81-x^2))\\\\ \boxed{\tan\theta=(x)/(√(81-x^2))}`