given tanθ= -2√10/3 and π/2<θ< π find:
sin2θ
tan θ/2
given sinθ=-5/13 and π<θ<3π/2 find
sin2θ
cos( θ-4π/3)
sin(θ/2)
can some1 explain this to me please?
Sure! I'd be happy to explain how to find the answers to these questions.
To find the values of trigonometric functions given an angle, we can use the properties and identities of trigonometric functions. Here's how we can solve each question step by step:
Question 1:
Given tan(θ) = -2√10/3 and π/2 < θ < π, we can find sin(2θ) and tan(θ/2).
1. To find sin(2θ), we can use the double angle identity for sine:
sin(2θ) = 2sin(θ)cos(θ).
2. To find tan(θ/2), we can use the half-angle identity for tangent:
tan(θ/2) = sin(θ) / (1 + cos(θ)).
3. Find θ: Since we know that π/2 < θ < π and that tan(θ) = -2√10/3, we can use the inverse tangent function to find the value of θ.
4. Substitute the value of θ into the equations from steps 1 and 2 to calculate sin(2θ) and tan(θ/2).
Question 2:
Given sin(θ) = -5/13 and π < θ < 3π/2, we can find sin(2θ), cos(θ - 4π/3), and sin(θ/2).
1. To find sin(2θ), we can use the double angle identity for sine:
sin(2θ) = 2sin(θ)cos(θ).
2. To find cos(θ - 4π/3), we can use the difference of angles formula for cosine:
cos(θ - 4π/3) = cos(θ)cos(4π/3) + sin(θ)sin(4π/3).
3. To find sin(θ/2), we can use the half-angle identity for sine:
sin(θ/2) = ±√((1 - cos(θ))/2). Since we are given π < θ < 3π/2, we know that sin(θ/2) will be negative.
4. Substitute the value of θ into the equations from steps 1, 2, and 3 to calculate sin(2θ), cos(θ - 4π/3), and sin(θ/2).
Remember to simplify the expressions using the values of θ you find.
I hope this explanation helps! Let me know if there's anything else I can assist you with.