In which quadrant does θ lie if the following statements are true: sin > θ
0 and cos θ >
In which quadrant does θ lie if the following statements are true: sin > θ
0 and tan θ >
Since sin > θ is positive in the first and second quadrants and tan θ > 0 is positive in the first and third quadrants, θ must lie in the first quadrant.
In which quadrant does θ lie if the following statements are true: tan < θ
0 and sin θ < 0
Since tan < θ is negative in the second and fourth quadrants and sin θ < 0 is negative in the third and fourth quadrants, θ must lie in the fourth quadrant.
Find the exact value of tan N in simplest form.
one side = √31
another √47
another 4
We can use the tangent formula to find the value of tan N:
tan N = opposite/adjacent
In this case, the opposite side is √47 and the adjacent side is 4. To find the value of the hypotenuse, we can use the Pythagorean theorem:
h^2 = (√31)^2 + (√47)^2 + 4^2
h^2 = 31 + 47 + 16
h^2 = 94 + 16
h^2 = 110
h = √110
Now we can plug in the values for the opposite, adjacent, and hypotenuse sides:
tan N = √47/4
To simplify, we can rationalize the denominator by multiplying both the numerator and denominator by √4:
tan N = (√47/4) x (√4/√4)
tan N = √188/4
tan N = √47/2
Therefore, the exact value of tan N in simplest form is √47/2.
Find the exact value of cot H in simplest form.
hypotenuse = √31
opposite to ∠N = √15
adjacent to ∠N = 4
We can use the definition of cotangent to find the value of cot H:
cot H = adjacent/opposite
In this case, the adjacent side is 4 and the opposite side is √15. To find the value of the hypotenuse, we can use the Pythagorean theorem:
h^2 = (√31)^2 + (√15)^2
h^2 = 31 + 15
h^2 = 46
h = √46
Now we can plug in the values for the adjacent, opposite, and hypotenuse sides:
cot H = 4/√15
To simplify, we can rationalize the denominator by multiplying both the numerator and denominator by √15:
cot H = (4/√15) x (√15/√15)
cot H = 4√15/15
Therefore, the exact value of cot H in simplest form is 4√15/15.
Given sin A 12/13 and that angle A is in Quadrant I, find the exact value of cotA in simplest radical form using a rational denominator.
We can use the formula for cotangent in terms of sine and cosine:
cot A = cos A / sin A
Since sin A = 12/13 and A is in Quadrant I, we can use the Pythagorean theorem to find the value of the adjacent side of the triangle:
cos A = √(1 - sin^2 A) = √(1 - (12/13)^2) = √(1 - 144/169) = √(25/169) = 5/13
Now we can plug in the values for sine and cosine:
cot A = cos A / sin A = (5/13) / (12/13) = 5/12
Therefore, the exact value of cotA in simplest radical form using a rational denominator is 5/12.