if sin θ = 2/3 and cos θ > 0, then find tan θ.

Question

*the '/' in '2/3', is indicating that it is a fraction, just to be a bit more specific*

Answer 1

The ratios of x, y, and their hypotenuse never change for a given angle. So if sin(θ) = y/hypotenuse = 2/3, we can use y=2 and 3=hypotenuse. Since we are given that θ is a right angle, we can use the Pythagorean theorem to find a value for x.

3×3 = 9 = x*x + 2*2 = x² + 4

Subtracting 4 from both sides,

5 = x²

Taking the square root of both sides,

√5 = x.

Now, tan(θ) = y / x = 2 / (√5).

NOTE: we were given that cos(θ) > 0, so x > 0.

Leaving a radical in the denominator is considered bad form, so let's rationalize the denominator:

[ 2 / (√5) ] * 1 = [ 2 / (√5) ] * [ (√5) / (√5) ] =

( 2*√5 ) / 5 = 0.4 * √5 ≈ 0.8944 to four decimal places.

Answer 2

cosθ >0 suggests that you will have to consider the one solution in the first quadrant. So, cosθ = +√1-sin²θ, from the relationship cos²θ + sin²θ = 1

tanθ = sinθ/cosθ = sinθ/(√1-sin²θ) = (2/3)/(√1-4/9) = (2/3)/√(5/9) = 2/√5

Answer 3

sinT = 2/3>0, cosT>0

so it's in the 1st quadrant

the only quadrant where sines, cosines and tangents are all positive

3 = hypotenuse of the reference angle

2 = opposite side

use the Pythagorean Theorem

adjacent side = sqr(3^2 -2^2) =sqr(9-4) = sqr5 ignore the negative square root

cosT =adjacent side/hypotenuse= sqr5/3

tanT = sinT/cosT = (2/3)/(sqr5/3)= 2/sqr5

= (2sqr5)/5= about .8944

T= about 41.81 degrees = sin^-1(2/3)= about .23pi or .73 radians

tan(41.81)= about .8944

2/sqr5= about .8944

Answer 4

I'll be typing my answer in code:

import math

# Given values

sin_theta = 2 / 3

cos_theta_positive = True

# Deduce the sides of the right-angled triangle

opposite_side = sin_theta

hypotenuse = 1

adjacent_side = math.sqrt(hypotenuse**2 - opposite_side**2)

# Check if cos(theta) is positive

if cos_theta_positive:

tan_theta = opposite_side / adjacent_side

else:

# Adjust the sign based on the quadrant

tan_theta = -opposite_side / adjacent_side

# Print the result

print(tan_theta)

The answer would be: 2 ⁄ √5