Answer:
- sin(θ) = -√21/5 ≈ -0.9165
- tan(θ) = -√21/2 ≈ -2.2913
Step-by-step explanation:
Given cos(θ) = 2/5 and π < θ < 2π, you want the sine and tangent of θ.
Quadrant
Angles are in the range π < θ < 2π in the 3rd and 4th quadrants. The cosine is positive in the 1st and 4th quadrants, so the desired angle is in the 4th quadrant. There, the sine and tangent are negative.
Pythagorean relation
The relationship between cos(θ) and sin(θ) is ...
sin(θ) = ±√(1 -cos²(θ))
sin(θ) = -√(1 -(2/5)²) = -√(1 -4/25)
sin(θ) = -(√21)/5 ≈ -0.9165
Tangent identity
The relation between the tangent and the sine and cosine is ...
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (-(√21)/5)/(2/5)
tan(θ) = -(√21)/2 ≈ -2.2913
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Additional comment
Once you know the sign of the trig functions you're looking for, a calculator can be used to find their numerical values easily. The exact values shown above are better found "by hand".
