Given that cos θ = 5/13 and θ is in QIV, find the exact value of
a. y
b. sin θ
c. sec θ
d. tan θ
If cosØ = 5/13 and Ø is in IV , then using the standard definition of cos Ø = x/r
we have x=5, r=13, then
5^2 + y^2 = 13^2
y^2 = 144
and in IV ....
a) y = -12
b) sinØ = -12/13
c) secØ = 13/5
d) tanØ = y/x = -12/5
To find the exact values of y, sin θ, sec θ, and tan θ, we can use the Pythagorean identity and the definitions of the trigonometric functions.
Given that cos θ = 5/13 and θ is in QIV, we know that cos θ is positive and sin θ is negative in QIV.
a. To find the value of y, we can use the Pythagorean identity:
sin^2 θ + cos^2 θ = 1
Since cos θ = 5/13, we can substitute the given value:
sin^2 θ + (5/13)^2 = 1
sin^2 θ + 25/169 = 1
sin^2 θ = 1 - 25/169
sin^2 θ = 144/169
Taking the square root of both sides, we get:
sin θ = ± 12/13
Since sin θ is negative in QIV, we take the negative value:
sin θ = -12/13
Now, we can use the definition of sine to find y:
sin θ = y / r
-12/13 = y / 13
y = -12
Therefore, the exact value of y is -12.
b. The exact value of sin θ is -12/13.
c. To find the value of sec θ, we can use the reciprocal identity:
sec θ = 1 / cos θ
sec θ = 1 / (5/13)
sec θ = 13/5
Therefore, the exact value of sec θ is 13/5.
d. To find the value of tan θ, we can use the definition of tangent:
tan θ = sin θ / cos θ
tan θ = (-12/13) / (5/13)
tan θ = -12/5
Therefore, the exact value of tan θ is -12/5.