Find the value of the remaining trigonometry functions if sec θ = 3/7 and θ ∈ IV.
Answer (1)
To find the values of the remaining trigonometry functions given sec θ = 7 3 and θ ∈ I V (Quadrant IV), let's follow these steps:
**Understand the Relationship: **
The secant function is the reciprocal of the cosine function, meaning sec θ = c o s θ 1 .
Therefore, we have cos θ = 3 7 .
**Evaluate the Sine and Tangent Functions: **
In Quadrant IV, cosine is positive and sine is negative.
Using the identity sin 2 θ + cos 2 θ = 1 , we can find sin θ : sin 2 θ = 1 − cos 2 θ = 1 − ( 3 7 ) 2 = 1 − 9 49 = 9 9 − 9 49 = − 9 40
Since sine is negative in Quadrant IV, sin θ = − − 9 40 . However, there's an error in computation as the square root can't be taken of a negative value here. Please check the initial conditions since they seem inconsistent.
**Find the Remaining Functions with Correct Values: **
Assuming a correction and continuing with reciprocal functions:
csc θ = s i n θ 1 .
tan θ = c o s θ s i n θ and cot θ = t a n θ 1 .
Conclusion:
With accurate inputs, use identities and quadrants for each function to check consistency, verify computations, and ensure conditions match real trigonometric scenarios.
Remember, in Quadrant IV:
cos θ is positive.
sin θ is negative.
Such initial conditions highlight a need for correction in assumptions or values given, always ensuring clarity and rechecking quadrants when needed.
