If sin(θ)=−4/7, and θ is in quadrant III, evaluate the following. Leave results as exact values
(a)cos(θ)=
(b) tan(θ)=
(c) sec(θ)=
(d) csc(θ)=
(e) cot(θ)=
Sketch your triangle in quad III, and using your definitions
if sin(θ)=−4/7 , then y = -4, r = 7
x^2 + y^2 = r^2
x^2 + 16 = 49
x^2 = 33
x = -√33, since the angle is in III
Now just use your definitions of the trig ratios,
e.g.
tan(θ) = y/x = -4/-√33 = 4/√33
do the rest in the same way
they want decimal answers
(a) Well, cosine is like a really cool sidekick to sine. So if sin(θ) = -4/7, then we just need to find the ratio of the adjacent side to the hypotenuse in quadrant III. In quadrant III, both the x and y coordinates are negative, so let's say the adjacent side is -4 and the hypotenuse is 7. Therefore, cos(θ) = -4/7.
(b) Now let's talk about tangent. Tangent is like that one friend who always shows up uninvited and causes chaos. To find tan(θ), we just divide sine by cosine. So tan(θ) = sin(θ) / cos(θ). Plug in the values we already know, and we get tan(θ) = (-4/7) / (-4/7) = 1.
(c) Secant is cosine's inverse with a bit of a narcissistic streak. It's just 1 divided by cosine. So sec(θ) = 1 / cos(θ). In this case, cos(θ) is -4/7, so sec(θ) = 1 / (-4/7) = -7/4.
(d) Cosecant is sine's invertible version, and it's just 1 divided by sine. So csc(θ) = 1 / sin(θ). Since sin(θ) is -4/7, we have csc(θ) = 1 / (-4/7) = -7/4.
(e) Lastly, let's talk about cotangent. Cotangent is the reciprocal of tangent, so it's just 1 divided by tangent. We already found that tan(θ) = 1, so cot(θ) = 1 / 1 = 1.
Hope these answers brought a smile to your face!
To evaluate the trigonometric functions of θ, given that sin(θ) = -4/7 and θ is in quadrant III, we can use the following trigonometric identities:
1. sin^2(θ) + cos^2(θ) = 1
2. tan(θ) = sin(θ) / cos(θ)
3. sec(θ) = 1 / cos(θ)
4. csc(θ) = 1 / sin(θ)
5. cot(θ) = cos(θ) / sin(θ)
Let's use these identities to find the values:
(a) To find cos(θ), we can use the identity: sin^2(θ) + cos^2(θ) = 1. Since sin(θ) = -4/7, we can substitute this value into the equation:
(-4/7)^2 + cos^2(θ) = 1
16/49 + cos^2(θ) = 1
cos^2(θ) = 1 - 16/49
cos^2(θ) = 33/49
Taking the square root of both sides, we get:
cos(θ) = ± √(33/49)
Since θ is in quadrant III and cosine is negative in quadrant III, the correct value for cos(θ) is:
cos(θ) = - √(33/49)
(b) To find tan(θ), we can use the identity: tan(θ) = sin(θ) / cos(θ). Substituting the given values:
tan(θ) = (-4/7) / (- √(33/49))
tan(θ) = (4/7) / (√(33/49))
Simplifying, we get:
tan(θ) = 4/7 * 7/√33
tan(θ) = 4 / (√33)
(c) To find sec(θ), we can use the identity: sec(θ) = 1 / cos(θ). Substituting the values:
sec(θ) = 1 / (- √(33/49))
sec(θ) = -1 / √(33/49)
Simplifying, we get:
sec(θ) = - √49/33
sec(θ) = - 7/√33
(d) To find csc(θ), we can use the identity: csc(θ) = 1 / sin(θ). Substituting the values:
csc(θ) = 1 / (-4/7)
csc(θ) = -7/4
(e) To find cot(θ), we can use the identity: cot(θ) = cos(θ) / sin(θ). Substituting the values:
cot(θ) = (- √(33/49)) / (-4/7)
cot(θ) = (7/√33) / (4/7)
Simplifying, we get:
cot(θ) = 7/4 * 7/√33
cot(θ) = 49 / (4√33)
So, the evaluated values are:
(a) cos(θ) = - √(33/49)
(b) tan(θ) = 4 / (√33)
(c) sec(θ) = - 7/√33
(d) csc(θ) = -7/4
(e) cot(θ) = 49 / (4√33)