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.
- Please help me with this question, I have no clue where to start as my teacher never taught us something like this!
cscθ cosθ = -1
1/sinθ cosθ = -1
cosθ/sinθ = -1
as with any angle,
cos^2θ + sin^2θ = 1
In this case,
cosθ = -sinθ
tanθ = -1
θ = 3π/4 or 7π/4
No idea at all? Hmmm. I cannot believe that the question was assigned, yet the material never taught!!!???
And have you never heard of google?
No problem! I can help you with this trigonometry problem step by step.
A) To express the left side of the given relation in terms of sin θ and cos θ, we can rewrite the reciprocal trigonometric ratios using the identities:
csc θ = 1/sin θ
cos θ = 1/sec θ
Substituting these values into the given relation, we get:
(1/sin θ) * (1/sec θ) = -1
Now, we can simplify this expression by finding the reciprocal of sec θ:
(1/sin θ) * (1/ (1/cos θ)) = -1
Simplifying further, we get:
(1/sin θ) * (cos θ/1) = -1
Now, we can combine the fractions:
cos θ / sin θ = -1
B) The relation between sin θ and cos θ can be found by using the trigonometric identity for tangent:
tan θ = sin θ / cos θ
From the given relation, we have:
cos θ / sin θ = -1
By rearranging this expression, we get:
sin θ / cos θ = -1
Comparing this with the identity for tangent, we can see that the relation is:
tan θ = -1
This means that the tangent of angle θ is equal to -1.
C) To determine two possible values for θ, we can look at the graph of the tangent function and find where its value is -1.
Since tangent is negative in quadrants II and IV, we need to find two angles in those quadrants where the tangent is -1.
In quadrant II, the reference angle for the tangent of -1 is π/4 radians (45 degrees). Adding π radians (180 degrees) gives us an angle of 5π/4 radians (225 degrees).
In quadrant IV, the reference angle for the tangent of -1 is 5π/4 radians (225 degrees). Adding 2π radians (360 degrees) gives us an angle of 13π/4 radians (405 degrees).
Therefore, two possible values for θ are 5π/4 radians (225 degrees) and 13π/4 radians (405 degrees).
I hope this explanation helps you understand how to approach this problem. Let me know if you have any further questions!