Answer:
sin 2Ф = -24/25
Explanation:
* Lets revise the trigonometry functions in the four quadrants
# First quadrant the measure of all angles is between 0° and 90°
∴ All the angles are acute
∴ All the trigonometry functions of any angle are positive
# Second quadrant the measure of all angles is between 90° and 180°
∴ All the angles are obtuse
∴ The value of sin of any angle is positive (cos and tan are negative)
# Third quadrant the measure of all angles is between 180° and 270°
∴ All the angles are reflex
∴ The value of tan of any angle is positive (sin and cos are negative)
# Fourth quadrant the measure of all angles is between 270° and 360°
∴ All the angles are reflex
∴ The value of cos any angle is positive ( sin and tan are negative)
* We will need to revise two identity to solve the question
# sin²Ф + cos²Ф = 1
# sin 2Ф = 2 sinФ cosФ
* Now lets solve the question
∵ cosФ = 3/5
∵ Ф is in the fourth quadrant
∴ The value of sinФ is negative
∵ sin²Ф + cos²Ф = 1
∴ sin²Ф + (3/5)² = 1
∴ sin²Ф + 9/25 = 1 ⇒ subtract 9/25 from both sides
∴ sin²Ф = 16/25 ⇒ take √ for both sides
∴ sinФ = ± 4/5
- We will chose the value -4/5 because Ф is in the fourth quadrant
∴ sinФ = -4/5
∵ sin 2Ф = 2 sinФ cosФ
∵ sinФ = -4/5 and cosФ = 3/5
∴ sin 2Ф = 2 (-4/5) (3/5) = -24/25