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 Q = 2sqrt6
a) cos Q
b) cotQ
c) cot(90 degrees - Q)
d) sin Q
Ok, finally , now it makes sense
And now it is easy.
given: secQ = 5, then cosQ = 1/5
So we have a right angles triangle, where the hypotenuse is 5, the side adjacent to angle Q is 1, and the opposite to angle Q is 2√6
Notice that the second function, tanQ = 2√6 was not necessary, since we can see that from our triangle.
a) cosQ = 1/5
b) cotQ = 1/tanQ = 1/(2√6)
c) cot(90 - Q) = tan Q = 2√6
d) sinQ = 2√6/5
SIN(90-Q)/COSEC(90-Q)=COS(90-Q)/COSEC(90-Q)
SIN(90-Q)/COSEC(90-Q)+COS(90-Q)/COSEC(90-Q)=1
To find the trigonometric functions using the given function values and trigonometric identities, we can use the following formulas:
1. cos^2Q + sin^2Q = 1
2. sec^2Q = 1 + tan^2Q
3. cotQ = 1/tanQ
Now, let's find the trigonometric functions step by step:
a) To find cosQ, we can use the identity cos^2Q + sin^2Q = 1. Rearrange the equation to solve for cosQ:
cos^2Q = 1 - sin^2Q
Since secQ = 5, we know that secQ = 1/cosQ. Substitute the value of secQ that is given:
1/cosQ = 5
Solving for cosQ:
cosQ = 1/5
b) To find cotQ, we can use the identity cotQ = 1/tanQ. Substitute the value of tanQ that is given:
cotQ = 1/(2√6)
Simplifying the expression:
cotQ = √6/12
c) To find cot(90 degrees - Q), we can use the identity cot(90 - Q) = tanQ. Substitute the value of tanQ that is given:
cot(90 - Q) = 2√6
d) To find sinQ, we can use the identity cos^2Q + sin^2Q = 1. Rearrange the equation to solve for sinQ:
sin^2Q = 1 - cos^2Q
Since secQ = 5, we can rewrite it as:
cosQ = 1/5
Substituting the value of cosQ in the equation:
sin^2Q = 1 - (1/5)^2
Simplifying the expression:
sin^2Q = 24/25
Taking the square root of both sides to find sinQ:
sinQ = ±√(24/25)
Please note that we take the ± because the sine function is positive in the first and second quadrants.
So, the final answers are:
a) cosQ = 1/5
b) cotQ = √6/12
c) cot(90 - Q) = 2√6
d) sinQ = ±√(24/25)