If sin of theta equals 3/8 and theta is in quadrant II. what are cos, tan, csc, cot, and sec of theta?

Question

If sin of theta equals 3/8 and theta is in quadrant II. what are cos, tan, csc, cot, and sec of theta?

2 Answers

Impact of this question

100952 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-14T23:43:34

See explanation below.

Explanation:

$sin θ = \frac{3}{8}$ $sintheta = 3/8$ , $θ$ $theta$ in quadrant $I I$ $II$

Imagine a right triangle being drawn on the cartesian plane, as in the following example.

enter image source here

Since sin = opposite/hypotenuse, the side opposite to $θ$ $theta$ is 3 and the hypotenuse is $8$ $8$ , we can rearrange our pythagorean theorem to find the adjacent side, b.

$a^{2} + b^{2} = c^{2}$ $a^2 + b^2 = c^2$

$b^{2} = c^{2} - a^{2}$ $b^2 = c^2 - a^2$

$b^{2} = 8^{2} - 3^{2}$ $b^2 = 8^2 - 3^2$

$b^{2} = 64 - 9$ $b^2 = 64 - 9$

$b^{2} = 55$ $b^2 = 55$

$b = - \sqrt{55}$ $b = -sqrt(55)$

So, now we know that the adjacent side measures $- \sqrt{55}$ $-sqrt(55)$ units (since the x axis is negative in quadrant II) in length. Thus we can deduce that $cos θ = - \frac{\sqrt{55}}{8}$ $costheta = -sqrt(55)/8$ and $tan θ = - \frac{3}{\sqrt{55}}$ $tantheta = -3/(sqrt(55))$

Now, for $csc θ, sec θ and cot θ$ $csctheta, sectheta and cottheta$ , we must apply the reciprocal identities.

$csc θ = \frac{1}{sin θ}$ $csctheta = 1/sintheta$

$sec θ = \frac{1}{cos θ}$ $sectheta = 1/costheta$

$cot θ = \frac{1}{tan θ}$ $cottheta = 1/tantheta$

Therefore, $csc θ = \frac{8}{3}, sec θ = - \frac{8}{\sqrt{55}} and cot θ = - \frac{\sqrt{55}}{3}$ $csctheta = 8/3, sectheta = -8/sqrt(55) and cottheta = -sqrt(55)/3$

To summarize, the six trigonometric ratios, we get:

$sin θ = \frac{3}{8}$ $sintheta = 3/8$

$cos θ = - \frac{\sqrt{55}}{8}$ $costheta = -sqrt(55)/8$

$tan θ = - \frac{3}{\sqrt{55}}$ $tantheta = -3/sqrt(55)$

$csc θ = \frac{8}{3}$ $csctheta = 8/3$

$sec θ = - \frac{8}{\sqrt{55}}$ $sectheta = -8/sqrt(55)$

$cot θ = - \frac{\sqrt{55}}{3}$ $cottheta = -sqrt(55)/3$

Hopefully this helps!

Answer 2 · 2017-05-24T09:16:49

$see explanation$ $"see explanation"$

Explanation:

$sin θ = \frac{3}{8} \to (1)$ $sintheta=3/8to(color(red)(1))$

$∙ csc θ = \frac{1}{sin θ} = \frac{8}{3} \to (2)$ $• csctheta=1/sintheta=8/3to(color(red)(2))$

$∙ cos θ = \pm \sqrt{1 - {sin}^{2} θ}$ $• costheta=+-sqrt(1-sin^2theta)$

$\times \times x = - \sqrt{1 - \frac{9}{64}} \leftarrow negative value$ $color(white)(xxxxx)=-sqrt(1-9/64)larr" negative value"$

$\times \times x = - \sqrt{\frac{55}{64}}$ $color(white)(xxxxx)=-sqrt(55/64)$

$\Rightarrow cos θ = - \frac{\sqrt{55}}{8} \to (3)$ $rArrcostheta=-sqrt55/8to(color(red)(3))$

$∙ sec θ = \frac{1}{cos θ} = - \frac{8}{\sqrt{55}} \to (4)$ $• sectheta=1/costheta=-8/sqrt55to(color(red)(4))$

$∙ tan θ = \frac{sin θ}{cos θ}$ $• tantheta=(sintheta)/(costheta)$

$\times \times = \frac{3}{8} \times - \frac{8}{\sqrt{55}} = - \frac{3}{\sqrt{55}} \to (5)$ $color(white)(xxxx)=3/8xx-8/sqrt55=-3/sqrt55to(color(red)(5))$

$∙ cot θ = \frac{1}{tan θ} = - \frac{\sqrt{55}}{3} \to (6)$ $• cottheta=1/tantheta=-sqrt55/3to(color(red)(6))$

Science

Math

Humanities

... and beyond

If sin of theta equals 3/8 and theta is in quadrant II. what are cos, tan, csc, cot, and sec of theta?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question