Chapter 7: Q. 39 (page 486)

if $\sin θ = \frac{1}{3}$ , is in quadrant 2 find exact value of
$a) \cos θ b) \sin (θ + \frac{π}{6}) c) \cos (θ - \frac{π}{3}) d) \tan (θ + \frac{π}{4})$

Short Answer

Expert verified

The values are

$a) \cos θ = - \frac{2 \sqrt{2}}{3} b) \sin (θ + \frac{π}{6}) = \frac{\sqrt{3} - 2 \sqrt{2}}{6} c) \cos (θ - \frac{π}{3}) = \frac{\sqrt{3} - 2 \sqrt{2}}{6} d) \tan (θ + \frac{π}{4}) = \frac{- 1 + 2 \sqrt{2}}{2 \sqrt{2} + 1}$

Step by step solution

Given information

We are given $\sin θ = \frac{1}{3}$

Find cosθ

We have,

$\cos θ = \sqrt{1 - \sin^{2} θ} \cos θ = \sqrt{1 - \frac{1}{9}} \cos θ = - \frac{2 \sqrt{2}}{3} a s θ i s i n 2 n d q u a d r a n t$

Find sin(θ+π6)

We have,

$\sin (θ + \frac{π}{6}) = \sin θ \cos \frac{π}{6} + \sin \frac{π}{6} \cos θ \sin (θ + \frac{π}{6}) = \frac{1}{3} (\frac{\sqrt{3}}{2}) + (- \frac{2 \sqrt{2}}{3}) (\frac{1}{2}) \sin (θ + \frac{π}{6}) = \frac{\sqrt{3} - 2 \sqrt{2}}{6}$

Find cos(θ+π3)

We have,

$\cos (θ - \frac{π}{3}) = \cos (θ) \cos (\frac{π}{3}) + \sin (θ) \sin (\frac{π}{3}) \cos (θ - \frac{π}{3}) = (- \frac{2 \sqrt{2}}{3}) \frac{1}{2} + \frac{1}{3} (\frac{\sqrt{3}}{2}) \cos (θ - \frac{π}{3}) = \frac{\sqrt{3} - 2 \sqrt{2}}{6}$

Find tan(θ+π4)

We have,

$\tan (θ + \frac{π}{4}) = \frac{\tan (θ) + \tan (\frac{π}{4})}{1 - \tan (θ) \tan (\frac{π}{4})} \tan (θ + \frac{π}{4}) = \frac{- \frac{1}{2 \sqrt{2}} + 1}{1 - (- \frac{1}{2 \sqrt{2}})} \tan (θ + \frac{π}{4}) = \frac{- 1 + 2 \sqrt{2}}{2 \sqrt{2} + 1}$

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if $\sin θ = \frac{1}{3}$ , is in quadrant 2 find exact value of
$a) \cos θ b) \sin (θ + \frac{π}{6}) c) \cos (θ - \frac{π}{3}) d) \tan (θ + \frac{π}{4})$

Short Answer

Step by step solution

Given information

Find cosθ

Find sin(θ+π6)

Find cos(θ+π3)

Find tan(θ+π4)

One App. One Place for Learning.

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if sinθ=13, is in quadrant 2 find exact value ofa)cosθb)sin(θ+π6)c)cos(θ-π3)d)tan(θ+π4)

Short Answer

Step by step solution

Given information

Find cosθ

Find sin(θ+π6)

Find cos(θ+π3)

Find tan(θ+π4)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Applied Mathematics

Geometry

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

if $\sin θ = \frac{1}{3}$ , is in quadrant 2 find exact value of
$a) \cos θ b) \sin (θ + \frac{π}{6}) c) \cos (θ - \frac{π}{3}) d) \tan (θ + \frac{π}{4})$