if tan(θ)=√(33) / 4 (√ does not ride over 4) find sin(θ)<0
i would just like to know the steps of how to do it.
To find sin(θ) when tan(θ) = √33/4, we can use the given information and the relationships between trigonometric functions to determine the sign of sin(θ).
Step 1: Recall the definition of tangent and sin in terms of the sides of a right triangle. Tangent is defined as the ratio of the opposite side (o) to the adjacent side (a), while sine is defined as the ratio of the opposite side (o) to the hypotenuse (h).
Step 2: Since tan(θ) = √33/4, we can assign values to the sides of a right triangle. Let the length of the opposite side be √33 and the length of the adjacent side be 4.
Step 3: Use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides. So, h^2 = (√33)^2 + 4^2.
Calculating this equation gives us h^2 = 33 + 16, which simplifies to h^2 = 49. Taking the square root of both sides, we have h = 7.
Step 4: Now that we know the lengths of all three sides, we can determine the sign of sin(θ). Since sine is defined as the ratio of the opposite side (o) to the hypotenuse (h), we can conclude that sin(θ) = √33/7.
Step 5: To determine the sign of sin(θ), we need to know the sign of √33. Since √33 is a positive number (assuming we are working with acute angles), sin(θ) will also be positive, meaning sin(θ) > 0.
Given that we are asked to find sin(θ) < 0, we can conclude that this condition is not satisfied for the given value of tan(θ).