PLEASE HELP!!
csc(theta)= -3/2
3pi/2 < theta < 2pi
Find:
sin(theta)= _____ ?
cos(theta)= _____ ?
tan(theta)= _____ ?
cot(theta)= _____ ?
sec(theta)= _____ ?
Knowing that cosecant is the reciprocal for the sin function, sinć =-2/3 because that is the reciprocal of
-3/2. Notice that it is also negative because reciprocal functions always share the same sign. For the
other functions, you will need to make use of the Pythagorean theorem. Since sin=
opposite/hypotenuse and the Pythagorean theorem is C2 = a2 + b2, we can substitute what we know.
for ¡¥a¡¦ we put 2, and ¡¥c¡¦ we put 3. Signs do not matter since they are being squared. This gives you
9 = 4 + b2 , and through a bit of math we get our adjacent side of the triangle, b= radical5, this is a +/- radical5,
Depending on which quadrant you are in and which function you are using to solve this will determine
the sign. In the fourth quadrant sine and tangent are negative, cosine is positive. Let¡¦s get the cosine and
secant functions values. Cosine is adjacent/hypotenuse and the secant is the reciprocal of that,
hypotenuse/ adjacent or rather 1 over adjacent/hypotenuse. Since we found the adjacent value to be
+/- rad5, cosć= rad5/3 and secć= 3/rad5. The tangent of an angle is the ratio of opposite/adjacent and the
cotangent is adjacent/opposite giving us tanć = -2/rad5 cotć =-rad5/2. Once you do these for a while the
functions become almost instinctual. Good Luck!
To find the values of sin(theta), cos(theta), tan(theta), cot(theta), and sec(theta) given that csc(theta) = -3/2 and theta is in the range 3pi/2 < theta < 2pi, we can use the following steps:
Step 1: Using the given information that csc(theta) = -3/2, we can determine the value of sin(theta). Since csc(theta) is the reciprocal of sin(theta), we can write:
sin(theta) = 1/csc(theta) = 1/(-3/2) = -2/3.
Therefore, sin(theta) = -2/3.
Step 2: To find cos(theta), we can use the identity:
sin^2(theta) + cos^2(theta) = 1.
Plugging in the value of sin(theta) we found in step 1, we can solve for cos(theta):
(-2/3)^2 + cos^2(theta) = 1,
4/9 + cos^2(theta) = 1,
cos^2(theta) = 1 - 4/9 = 5/9.
Taking the square root of both sides, we get:
cos(theta) = ± sqrt(5/9).
Since theta is in the given range, 3pi/2 < theta < 2pi, cosine is positive in the fourth quadrant. Thus:
cos(theta) = sqrt(5/9) = sqrt(5)/3.
Step 3: To find tan(theta), we can use the identities:
tan(theta) = sin(theta)/cos(theta).
Plugging in the values of sin(theta) and cos(theta) we found in steps 1 and 2, respectively, we get:
tan(theta) = (-2/3) / (sqrt(5)/3) = -2/sqrt(5).
Step 4: To find cot(theta), we can use the identity:
cot(theta) = 1/tan(theta).
Plugging in the value of tan(theta) we found in step 3, we can simplify it to:
cot(theta) = 1 / (-2/sqrt(5)) = -sqrt(5)/2.
Step 5: To find sec(theta), we can use the identity:
sec(theta) = 1/cos(theta).
Plugging in the value of cos(theta) we found in step 2, we get:
sec(theta) = 1/sqrt(5/9) = sqrt(9/5) = 3/sqrt(5).
Therefore, the values of sin(theta), cos(theta), tan(theta), cot(theta), and sec(theta) are:
sin(theta) = -2/3,
cos(theta) = sqrt(5)/3,
tan(theta) = -2/sqrt(5),
cot(theta) = -sqrt(5)/2,
sec(theta) = 3/sqrt(5).
To solve the given problem, we need to find the values of the trigonometric functions sin(theta), cos(theta), tan(theta), cot(theta), and sec(theta) when csc(theta) is given as -3/2.
First, let's identify which quadrant theta lies in based on the given range 3pi/2 < theta < 2pi.
The range 3pi/2 < theta < 2pi corresponds to the fourth quadrant in the coordinate plane.
In the fourth quadrant, the sine function is negative, while the cosine function is positive.
1. sin(theta):
To find sin(theta), we can use the relationship between sin(theta) and csc(theta):
sin(theta) = 1 / csc(theta)
Therefore, sin(theta) = 1 / (-3/2) = -2/3
2. cos(theta):
To find cos(theta), we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
From the previous step, we know sin(theta) = -2/3. Substituting this into the equation, we can solve for cos(theta):
(-2/3)^2 + cos^2(theta) = 1
4/9 + cos^2(theta) = 1
cos^2(theta) = 1 - 4/9
cos^2(theta) = 5/9
cos(theta) = ±√(5/9)
Since theta is in the fourth quadrant, cos(theta) is positive. Therefore, cos(theta) = √(5/9) = √5/3
3. tan(theta):
To find tan(theta), we can use the relationship between tan(theta) and sin(theta) and cos(theta):
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (-2/3) / (√5/3)
tan(theta) = -2/√5 * 3/3
tan(theta) = -2/√5 = -2√5/5
4. cot(theta):
To find cot(theta), we can use the relationship between cot(theta) and tan(theta):
cot(theta) = 1 / tan(theta)
cot(theta) = 1 / (-2√5/5)
cot(theta) = -5/2√5 = -5√5/10 = -√5/2
5. sec(theta):
To find sec(theta), we can use the relationship between sec(theta) and cos(theta):
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (√5/3)
sec(theta) = 3/√5 * √5/√5
sec(theta) = 3/5√5 = 3√5/25
Therefore, the values of the trigonometric functions are:
sin(theta) = -2/3
cos(theta) = √5/3
tan(theta) = -2√5/5
cot(theta) = -√5/2
sec(theta) = 3√5/25