Find the remaining trigonometric values if sin=-3/5 and theta terminates in Quadrent four.
I'm not sure what they're asking
For any given angle, there are 6 trig ratios.
They gave you one of them.
sin Ø = -3/5 --- (you have a typo there)
you should know that sinØ = y/r, and r is always positive.
so you have y = -3, r = 5
by Pythagoras, x^2 + y^2 = r^2
x^2 + 9 = 25
x^2 = 16
x = ± 4, but we are told that the angle is in IV,
so x = +4
sinØ = -3/5 ---> that was given
cscØ = -5/3
cosØ = x/r = 4/5
secØ = 5/4
tanØ = y/x = -3/5
cotØ = -5/3
theta is a Quad IV angle with a sine of -3/5
find cos, tan, sec, csc and cot
To find the remaining trigonometric values, we first need to determine the values of other trigonometric functions.
Given that sin(theta) = -3/5 and theta terminates in Quadrant four, we can use the following steps to find the values of other trigonometric functions:
1. Start by drawing a right triangle in Quadrant four, labeled with an angle theta.
2. Since sin(theta) = opposite/hypotenuse, we can let the opposite side be -3 and the hypotenuse be 5 (using the given values).
3. To find the adjacent side, we can use the Pythagorean theorem: a^2 + b^2 = c^2. In this case, a represents the adjacent side, b represents the opposite side, and c represents the hypotenuse. Thus, a^2 + (-3)^2 = 5^2. Solving this equation, we get a^2 = 25 - 9, which gives a^2 = 16. Taking the square root of both sides, we find that a = +/- 4. However, since theta terminates in Quadrant four, the adjacent side should be positive. Therefore, a = 4.
4. Now, we can find the values of other trigonometric functions:
- cos(theta) = adjacent/hypotenuse = 4/5
- tan(theta) = opposite/adjacent = -3/4
- csc(theta) = 1/sin(theta) = 1/(-3/5) = -5/3
- sec(theta) = 1/cos(theta) = 1/(4/5) = 5/4
- cot(theta) = 1/tan(theta) = 1/(-3/4) = -4/3
So, the remaining trigonometric values are:
- cos(theta) = 4/5
- tan(theta) = -3/4
- csc(theta) = -5/3
- sec(theta) = 5/4
- cot(theta) = -4/3
The question is asking for the remaining trigonometric values of an angle (theta) in the fourth quadrant given that the sine of theta is -3/5.
To solve this problem, we need to recall the definitions of the trigonometric functions in the fourth quadrant. In the fourth quadrant, both x and y coordinates are negative. Using the given information that sin(theta) = -3/5, we can determine the values of the other trigonometric functions using the Pythagorean identity and the definitions of these functions.
1. Recall that sin(theta) = y/r, where y represents the length of the side opposite to angle theta, and r represents the hypotenuse (the radius of the unit circle). Since sin(theta) = -3/5, we can determine that y = -3 and r = 5.
2. Now, we can find the value of x, which represents the length of the side adjacent to angle theta, using the Pythagorean identity: x^2 + y^2 = r^2. Substitute the values of y and r we found earlier: x^2 + (-3)^2 = 5^2. Simplifying the equation gives x^2 = 25 - 9 = 16. Hence, x = ±4.
Now that we have the values of x, y, and r, we can determine the remaining trigonometric values in the fourth quadrant:
1. Cosine (cos): In the fourth quadrant, cosine is positive. Since cos(theta) = x/r, substitute the values of x and r we found earlier: cos(theta) = 4/5.
2. Tangent (tan): In the fourth quadrant, tangent is negative. Since tan(theta) = y/x, substitute the values of y and x we found earlier: tan(theta) = -3/4.
3. Cosecant (csc): Since csc(theta) = 1/sin(theta), substitute the given value of sin(theta): csc(theta) = 1/(-3/5) = -5/3.
4. Secant (sec): Since sec(theta) = 1/cos(theta), substitute the value of cos(theta): sec(theta) = 1/(4/5) = 5/4.
5. Cotangent (cot): Since cot(theta) = 1/tan(theta), substitute the value of tan(theta): cot(theta) = 1/(-3/4) = -4/3.
Therefore, the remaining trigonometric values for an angle in the fourth quadrant with sin = -3/5 are:
cos(theta) = 4/5
tan(theta) = -3/4
csc(theta) = -5/3
sec(theta) = 5/4
cot(theta) = -4/3.