SOLUTION:
Case: Trigonometry (Quadrants)
Given:
[tex]\begin{gathered} sinA=\text{ -}\frac{4}{5} \\ In\text{ quadrant 3} \end{gathered}[/tex]Required:
Find sin (2A)
Method:
Step 1: First we find the acute angle A, that sinA = 4/5
[tex]\begin{gathered} sinA=\frac{\text{ 4}}{5} \\ A=\text{ }\sin^{-1}(\frac{4}{5}) \\ A=\text{ 53.13} \end{gathered}[/tex]From here,
CosA is:
[tex]\begin{gathered} cosA=\text{ }\frac{adj}{hyp} \\ using\text{ the 3-4-5 pythagoras rule, adj=3} \\ cosA=\text{ }\frac{3}{5} \end{gathered}[/tex]Step 2: Rotate the angle into the 3rd quadrant
A*= 53.13 + 180
A*= 233.13.
Step 3: Sin (2A)
[tex]\begin{gathered} sin(2A)=\text{ 2sinAcosA} \\ sin(2A)=\text{ 2}\times\text{\lparen}\frac{-4}{5}\text{\rparen}\times(\frac{\text{-3}}{5}\text{\rparen} \\ sin(2A)=\frac{\text{ 24}}{25} \end{gathered}[/tex]Final answer:
The value of sin(2A)= 24/25
