Given that sin 64° = 9/10 and cos 64° = (√19)/10 find the following:
1. cot 64° =
2. cos -64° =
3. csc 64° =
4. csc -26° =
5. sec 244° =
since sin = y/r and cos = x/r, you have
x = √19
y = 9
r = 10
Now look at the other functions.
tan = y/x
sec = r/x
etc.
are these correct?
1. (√19)/9
2. √19/10
3. 1/10
and i'm not sure how to do 4 and 5
3. csc 64° = = r/y = 10/√19
4. csc -26° = -csc(26°) = -sec(64°) = -r/x = -10/√19
5. sec 244° = sec(180°+64°) = -sec(64°) = -r/x = -10/√19
For #4, recall that 64° and 26° are complementary angles. So, swap func and co-func. That's what the co- means
cosine(x) = sine(co-x)
To find the values of the trigonometric ratios, we can use the relationships between trigonometric functions on the unit circle. Let's go through each question one by one.
1. To find cot 64°, we can use the relationship: cotθ = 1/tanθ. Since we are given sin 64° and cos 64°, we can use these values to find the value of tan 64° using the formula: tanθ = sinθ / cosθ.
First, sin 64° = 9/10, and cos 64° = (√19)/10.
Next, we can substitute these values into the formula:
tan 64° = sin 64° / cos 64° = (9/10) / (√19/10) = 9/√19
Finally, we can find cot 64° using the reciprocal property of tangents:
cot 64° = 1 / tan 64° = 1 / (9/√19) = √19/9
So, cot 64° = √19/9.
2. To find cos -64°, we can use the property: cos(-θ) = cos(θ). Therefore, cos -64° = cos 64° = (√19)/10.
So, cos -64° = (√19)/10.
3. To find csc 64°, we can use the relationship: cscθ = 1/sinθ. Since we are given sin 64°, we can use this value to find the value of csc 64° using the formula: csc 64° = 1 / sin 64°.
Given sin 64° = 9/10, we can substitute this value into the formula:
csc 64° = 1 / (9/10) = 10/9
So, csc 64° = 10/9.
4. To find csc -26°, we can use the property: csc(-θ) = -csc(θ). Therefore, csc -26° = -csc 26°.
To find csc 26°, we need the value of sin 26°. Since we are not given this value, we cannot find csc 26° or csc -26°.
5. To find sec 244°, we can use the relationship: secθ = 1/cosθ. Since we are given cos 244°, we can use this value to find the value of sec 244° using the formula: sec 244° = 1 / cos 244°.
Given cos 244° = (√19)/10, we can substitute this value into the formula:
sec 244° = 1 / ((√19)/10) = 10 / √19 = (10√19) / 19
So, sec 244° = (10√19) / 19.