use even and odd properties of the trigonometric functions to find the exact value of the expression.
sin(−π/6)
sin(−π/6)
-π/6 is in quadrant IV, in that quadrant the sine value is negative.
the angle in standard position is π/6
so sin(−π/6) = - sin π/6
you should know the ratio of sides of the standard 30-60-90 triangle
and thus
sin(−π/6) = -1/2
using even an odd properties, we get
sin (-x) = -sin x , which lead to the same steps above
for
cos(-x) = cos x , (which does not apply to this problem)
To find the exact value of sin(−π/6) using the even and odd properties of trigonometric functions, we can break it down step by step:
Step 1: Recall the even and odd properties of trigonometric functions:
- The even property states that sin(-θ) = -sin(θ) and cos(-θ) = cos(θ).
- The odd property states that tan(-θ) = -tan(θ), csc(-θ) = -csc(θ), sec(-θ) = sec(θ), and cot(-θ) = -cot(θ).
Step 2: Substitute the given angle into the even property of sin:
sin(-θ) = -sin(θ)
Step 3: Substitute the value of θ = π/6 into the equation derived in Step 2:
sin(-π/6) = -sin(π/6)
Step 4: Evaluate the right-hand side of the equation:
sin(π/6) is a commonly known value. It is equal to 1/2.
Step 5: Substitute the value of sin(π/6) into the equation derived in Step 3:
sin(-π/6) = -1/2
Therefore, the exact value of sin(−π/6) is -1/2.
To find the exact value of sin(−π/6) using the even and odd properties of trigonometric functions, we first need to know the values of sin(π/6).
The reference angle for π/6 is π/6 itself since it is positive and lies within the first quadrant. The value of sin(π/6) is 1/2.
Now, we can use the even property of the sine function, which states that sin(−θ) = −sin(θ), to find the value of sin(−π/6).
Applying the even property, we have:
sin(−π/6) = −sin(π/6)
Since sin(π/6) = 1/2, we can substitute it into the equation:
sin(−π/6) = −(1/2)
Therefore, the exact value of sin(−π/6) is −1/2.