Determine the exact primary trig ratios of the angle 240 degrees.
help
240 = 180 + 60
So the "related" angle to 240° is 60°
240° is in quadrant III, where only the tangent is positive.
so
sin240° = - sin60° = -√3/2
csc 240° = -2/√3
cos 240 = -cos60° = - 1/2
sec 240° = -2
tan 240° = √3
cot 240° = 1/√3
Your should know the basic side ratios, thus the trig ratios,
of the 30-60-90 ° and the 45-45 90 ° triangles.
That is how I got those answers
thank you ! this helped a lot!
To determine the exact primary trigonometric ratios of an angle, you can use the unit circle.
1. Start by converting the angle from degrees to radians. Since there are 2π radians in a full circle (360 degrees), you can convert 240 degrees to radians by multiplying it by π/180:
240 degrees × π/180 = 4π/3 radians
2. Place the angle on the unit circle. The angle of 4π/3 radians falls on the lower left quadrant.
3. Identify the coordinates of the point where the angle intersects the unit circle. In this case, the point lies on the unit circle with radius 1 and an angle of 4π/3 radians. The coordinates of this point are approximately (-0.5, -√3/2).
4. Now, you can determine the primary trigonometric ratios:
- Sine (sin) is the y-coordinate of the point: sin(240°) = sin(4π/3) ≈ -√3/2
- Cosine (cos) is the x-coordinate of the point: cos(240°) = cos(4π/3) ≈ -0.5
- Tangent (tan) is sin divided by cos: tan(240°) = tan(4π/3) ≈ (√3/2) / (-0.5)
So, the exact primary trigonometric ratios of the angle 240 degrees are:
- sin(240°) ≈ -√3/2
- cos(240°) ≈ -0.5
- tan(240°) ≈ (√3/2) / (-0.5)
To determine the exact primary trigonometric ratios of an angle, we will first convert the given angle to a reference angle within the range of 0 to 90 degrees.
Since 240 degrees is greater than 180 degrees, we subtract 180 from it:
240 - 180 = 60 degrees
Now, we have an angle of 60 degrees, which is within the first quadrant. In the first quadrant, the primary trigonometric ratios are as follows:
- Sine (sin): Opposite/Hypotenuse
- Cosine (cos): Adjacent/Hypotenuse
- Tangent (tan): Opposite/Adjacent
Using the reference angle of 60 degrees, we can construct a right triangle in the first quadrant, where the opposite side is represented by "y", the adjacent side is represented by "x", and the hypotenuse is represented by "r".
Now we can apply the ratios:
1. Sine of 60 degrees:
sin(60) = Opposite/Hypotenuse
= y / r
2. Cosine of 60 degrees:
cos(60) = Adjacent/Hypotenuse
= x / r
3. Tangent of 60 degrees:
tan(60) = Opposite/Adjacent
= y / x
To determine the exact values of these ratios, we need to use the properties of special right triangles.
In a 30-60-90 triangle, the ratio of the sides is:
- Opposite/Hypotenuse = √3/2
- Adjacent/Hypotenuse = 1/2
- Opposite/Adjacent = √3
Since we have a 60-degree angle, which is in a 30-60-90 triangle, we can substitute these values into the ratios we derived earlier:
1. Sine of 60 degrees:
sin(60) = y / r
= √3/2
2. Cosine of 60 degrees:
cos(60) = x / r
= 1/2
3. Tangent of 60 degrees:
tan(60) = y / x
= √3
Therefore, the exact primary trigonometric ratios for an angle of 240 degrees are as follows:
- Sine (sin): √3/2
- Cosine (cos): 1/2
- Tangent (tan): √3