Use the given function values and trigonometric identities (including the relationship between a trigonometric function and its cofunction of a complementary angle) to find the indicated trigonometric functions.
sec Q = 5 tan = 2sqrt6
a) cos Q b) cotQ
c) cot(90 degrees - Q) d) sin Q
Please explain. I do not understand how to do this. Thank you!!
sec Q = 5 tanQ
1/cosQ = 5sinQ/cosQ
so sinQ = 1/5
I then made a right angles triangle with hypotenuse 5, opposite 1 and let x be the adjacent.
By Pythagoras
x^2 + 1^2 = 5^2
x^2 = 24
x = √24 = 2√6
WOAH!
but you said secQ = 2√6
if, as you started, secQ = 5tanQ, then my calculations stands as correct and
secQ would be 5/(2√6) and not 2√6
I will wait till you correct your opening statement of
sec Q = 5 tan = 2sqrt6 , which leads to a contradiction.
What this person means is :
cosQ = 5 and tanQ = 2sqrt6
To find the trigonometric functions of angle Q, we can use the given function values and trigonometric identities.
a) cos Q:
Since sec Q = 1/cos Q, we can find cos Q by taking the reciprocal of sec Q. In this case, sec Q = 5, so:
cos Q = 1/5
b) cot Q:
Since tan Q = sin Q/ cos Q, we can find cot Q by taking the reciprocal of tan Q. In this case, tan Q = 2√6, so:
cot Q = 1/ tan Q = 1/(2√6)
c) cot(90 degrees - Q):
The trigonometric identity for cotangent states that cot(90 degrees - A) = tan A. Therefore, cot(90 degrees - Q) is equal to tan Q. We can use the given value of tan Q to directly find cot(90 degrees - Q):
cot(90 degrees - Q) = cot(90 degrees - Q) = tan Q = 2√6
d) sin Q:
Since sec Q = 1/cos Q, and the identity sin^2 A + cos^2 A = 1, we can find sin Q using these two identities.
Given that sec Q = 5, we can use the identity sec^2 A - 1 = tan^2 A to find cos Q:
sec^2 Q = 5^2 = 25
cos^2 Q = 1 - (1/sec^2 Q) = 1 - (1/25) = 24/25
Since sin^2 Q + cos^2 Q = 1, we can find sin Q:
sin^2 Q = 1 - cos^2 Q = 1 - (24/25) = 1/25
Taking the square root, we find:
sin Q = √(1/25) = 1/5
So, the values of the trigonometric functions are:
a) cos Q = 1/5
b) cot Q = 1/(2√6)
c) cot(90 degrees - Q) = tan Q = 2√6
d) sin Q = 1/5