An angle θ satisfies the relation csc θ cos θ = -1.
A) Use the definition of the reciprocal trigonometric ratios to express the left side in terms of sin θ and cos θ.
B) What is the relation between sin θ and cos θ for this angle?
C) Determine two possible values for θ.
cos /sin = -1 = cotan
90 + 45
360 - 45
A) To express the left side of the equation in terms of sin θ and cos θ, we need to use the definition of the reciprocal trigonometric ratios.
Recall that the reciprocal of sin θ is csc θ, and the reciprocal of cos θ is sec θ. Therefore, we can rewrite csc θ cos θ as 1/sin θ * cos θ.
B) Now, we need to determine the relation between sin θ and cos θ for this angle.
Since the given equation is csc θ cos θ = -1, we can substitute the expression we obtained in part A:
1/sin θ * cos θ = -1
Multiplying both sides of the equation by sin θ, we get:
cos θ = -sin θ
This implies that the relation between sin θ and cos θ is that they are negative reciprocals of each other. In other words, if sin θ = x, then cos θ = -1/x, and vice versa.
C) To determine possible values for θ, we can use the unit circle or a calculator to find angles where sin θ = x and cos θ = -1/x.
Let's consider x = 1/2 as an example. To find one possible value for θ, you can use a calculator or table of trigonometric values and look for an angle whose sine is 1/2. One such angle is θ = π/6 (30 degrees).
Substituting θ = π/6 into the equation csc θ cos θ = -1:
csc (π/6) * cos (π/6) = -1
(2/√3) * (√3/2) = -1
1 = -1
The equation is not satisfied for this value of θ.
To find another possible value for θ, we can consider x = -1/2. Again, using a calculator or table of trigonometric values, we find an angle whose sine is -1/2. One such angle is θ = 7π/6 (210 degrees).
Substituting θ = 7π/6 into the equation csc θ cos θ = -1:
csc (7π/6) * cos (7π/6) = -1
(-2/√3) * (-√3/2) = -1
1 = -1
The equation is satisfied for this value of θ.
Therefore, two possible values for θ that satisfy the given equation are θ = π/6 (30 degrees) and θ = 7π/6 (210 degrees).