Trigonometric Functions and Properties Quiz

The sine of an angle is the ratio of the opposite side to the hypotenuse. In this case, the opposite side is 6 and the hypotenuse is 10. Finding the hypotenuse using the Pythagorean Theorem gives us a hypotenuse of 10. Therefore, sin θ = 6/10, which simplifies to 3/5.

The quadrant of θ is determined by the signs of sinθ and cosθ. Since sinθ is negative, we know that the y-coordinate of the point on the unit circle is negative. Since secθ is positive, we know that cosθ is positive, meaning the x-coordinate of the point on the unit circle is positive. This means the point on the unit circle lies in quadrant IV.

We know that tanθ = sinθ/cosθ, and we're given that tanθ= 5/4 and sinθ>0. Since both the tangent and sine are positive, the angle must be in quadrant 1. In quadrant 1, cosine is also positive. To find cosθ, we can use the Pythagorean identity: sin²θ + cos²θ = 1. Solving for cosθ gives us cos θ = 4/5.

The reference angle is the smallest angle between the terminal side of θ and the x-axis. To find the reference angle, we can subtract the nearest multiple of π from the given angle. In this case, the nearest multiple of π is π, and subtracting it from 11π/6 gives us π/6. Therefore, the reference angle of 11π/6 is π/6.

The exact value of sin(7π/6)=-1/2. The sine function is negative in the third quadrant, where 7π/6 lies. The reference angle for 7π/6 is π/6, and the sine of π/6 is 1/2, so the sine of 7π/6 is -1/2.