Given that sec 3π/10 ≈ 17/10 and csc 3π/10 ≈ 17/14, find the following:
1. sin 3π/10 ≈ 14/17 (is this one correct)
2. csc 43π/10 ≈
3. sec 2π/10 ≈
4. cot -12π/10 ≈
5. tan π/5 ≈
6. sin -7π/10 ≈
7. cos 13π/10 ≈
8. tan -33π/10 ≈
i don't understand how to do the others can someone please explain?
Didn't we just do one of these?
sec 3π/10 = 17/10, csc 3π/10 = 17/14 means
r=17
x=10
y=14
csc 43π/10 = csc(4π + 3π/10) = csc 3π/10
sec 2π/10 = sec(π/2 - 3π/10) = csc 3π/10
cot -12π/10 = -cot 12π/10 = -cot(π+2π/10) = -cot 2π/10 = -tan 3π/10
Try the others. Recall the basic definitions of the trig functions in terms of x,y,r.
To find the values of trigonometric functions using the given approximations, we can use the reciprocal identities and the unit circle to determine the values step by step. Let's solve the remaining questions:
2. To find csc(43π/10), we can use the reciprocal identity: csc(θ) = 1/sin(θ). Since we know the value of csc(3π/10), which is approximately equal to 17/14, we can replace it in the equation as follows:
csc(43π/10) = 1/sin(43π/10) = 1/(sin(3π/10)) ≈ 1/((17/10)) = 10/17.
3. To find sec(2π/10), we can use the reciprocal identity: sec(θ) = 1/cos(θ). Since we know the value of sec(3π/10), which is approximately equal to 17/10, we can replace it in the equation as follows:
sec(2π/10) = 1/cos(2π/10) = 1/(cos(3π/10)) ≈ 1/(17/10) = 10/17.
4. To find cot(-12π/10), we can use the reciprocal identity: cot(θ) = 1/tan(θ). We need to find the value of tan(-12π/10). The tangent function has a period of π, meaning tan(θ) = tan(θ + π). Therefore, we can rewrite tan(-12π/10) as tan(2π/10):
tan(2π/10) = sin(2π/10) / cos(2π/10). Now we can use the previously given approximations:
sin(3π/10) ≈ 17/10, and cos(3π/10) = 1/sin(3π/10) ≈ 10/17.
sin(2π/10) = sin(π/5) ≈ sin(3π/10) ≈ 17/10.
cos(2π/10) = cos(π/5) ≈ cos(3π/10) ≈ 10/17.
Thus, tan(2π/10) ≈ (17/10) / (10/17) = 289/100 ≈ 2.89.
Finally, cot(-12π/10) = 1/tan(-12π/10) = 1/(tan(2π/10)) ≈ 1/2.89 ≈ 0.346.
5. To find tan(π/5), we can use the tangent identity: tan(θ) = sin(θ) / cos(θ). As mentioned earlier, sin(3π/10) ≈ 17/10, and cos(3π/10) ≈ 10/17.
Thus, tan(π/5) ≈ (17/10) / (10/17) = 289/100 ≈ 2.89.
6. To find sin(-7π/10), we can use the fact that sine is an odd function, meaning sin(-θ) = -sin(θ):
sin(-7π/10) = -sin(7π/10) = -(sin(3π/10)) ≈ -(17/10) = -17/10.
7. To find cos(13π/10), we can again use the fact that cosine is an even function, meaning cos(-θ) = cos(θ):
cos(13π/10) = cos(-13π/10) = cos(3π/10) ≈ 10/17.
8. To find tan(-33π/10), we can use the fact that tangent is an odd function, meaning tan(-θ) = -tan(θ):
tan(-33π/10) = -tan(33π/10) = -(tan(π/10)) ≈ -(14/17) = -14/17.
Please note that the approximations might introduce some errors in the calculations, so the final answers may not be exact.