In triangle GHI, angle H is a right angle, GH = 40, and cos G= 40/41. Draw a diagram and find each value in fraction and in decimal form.
a) Sin G
b. Sin I
c. Cot G
d. csc G
e. cos I
f. sec H
- I have no idea how to do this
the triangle's 3rd side is √(41^2-40^2) = 9
So, now you know all three sides, and the angles, so the trig functions are easy to read off.
sin G = 9/41
and so on
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To solve this problem, we will use the given information to find the values step-by-step.
Step 1: Draw the triangle GHI
Start by drawing a right-angled triangle GHI, with the right angle at H.
G
/|
/ |
/ |
I-----H
Step 2: Find the missing side of the triangle
Since GH is given as 40, we need to find the lengths of GI and HI.
Using the Pythagorean theorem:
GH^2 + HI^2 = GI^2
(40)^2 + HI^2 = GI^2
1600 + HI^2 = GI^2
Since angle H is a right angle, GI is equal to HI.
HI^2 + HI^2 = GI^2
2(HI^2) = GI^2
2(HI^2) = 1600
HI^2 = 800
Therefore, HI = sqrt(800) ≈ 28.28
Step 3: Find sin G
To find sin G, we use the side opposite angle G (HI) and the hypotenuse (GI).
sin G = opposite / hypotenuse
sin G = HI / GI
sin G = 28.28 / 40
In decimal form, sin G ≈ 0.7071
In fraction form, sin G ≈ 14√2 / 20
Step 4: Find sin I
To find sin I, we use the side opposite angle I (GH) and the hypotenuse (GI).
sin I = GH / GI
sin I = 40 / 40 (since GH = GI)
In decimal form, sin I = 1
In fraction form, sin I = 1/1
Step 5: Find cot G
To find cot G, we use the adjacent side (HI) and the opposite side (GH) of angle G.
cot G = adjacent / opposite
cot G = HI / GH
cot G = 28.28 / 40
In decimal form, cot G ≈ 0.7071
In fraction form, cot G ≈ 14√2 / 20
Step 6: Find csc G
To find csc G, we use the hypotenuse (GI) and the opposite side (HI) of angle G.
csc G = hypotenuse / opposite
csc G = GI / HI
csc G = 40 / 28.28
In decimal form, csc G ≈ 1.4142
In fraction form, csc G ≈ 20√2 / 14
Step 7: Find cos I
To find cos I, we use the adjacent side (GH) and the hypotenuse (GI) of angle I.
cos I = adjacent / hypotenuse
cos I = GH / GI
cos I = 40 / 40 (since GH = GI)
In decimal form, cos I = 1
In fraction form, cos I = 1/1
Step 8: Find sec H
To find sec H, we use the hypotenuse (GI) and the adjacent side (GH) of angle H.
sec H = hypotenuse / adjacent
sec H = GI / GH
sec H = 40 / 40 (since GH = GI)
In decimal form, sec H = 1
In fraction form, sec H = 1/1
Summary:
a) sin G ≈ 0.7071 (or 14√2 / 20)
b) sin I = 1 (or 1/1)
c) cot G ≈ 0.7071 (or 14√2 / 20)
d) csc G ≈ 1.4142 (or 20√2 / 14)
e) cos I = 1 (or 1/1)
f) sec H = 1 (or 1/1)
No problem! I will guide you through the process step by step.
Step 1: Draw a diagram:
Visualize a triangle GHI. Make sure that angle H is a right angle (90 degrees) and GH is 40 units long.
G
/|
/ |
/ |
/ |
/____H
I
Step 2: Find the missing side:
Since GH is given as 40 units and we are given the cosine of angle G, we can use the cosine ratio to find HI, the missing side.
The cosine ratio is defined as cos(angle) = adjacent side / hypotenuse.
In this case, we have cos(G) = GH / HI.
Substituting the values we know, cos(G) = 40/41, and GH = 40, we can solve for HI.
cos(G) = GH / HI
40/41 = 40 / HI
Cross-multiply and solve for HI:
40 * HI = 40 * 41
HI = 41
So now we know that HI is equal to 41 units.
Step 3: Calculate the remaining values:
a) Sin G:
The sine ratio is defined as sin(angle) = opposite side / hypotenuse.
In this case, we have G as angle G, HI as the hypotenuse, and GH as the opposite side.
sin(G) = GH / HI
sin(G) = 40 / 41
So the value of sin(G) is 40/41 or approximately 0.9756.
b) Sin I:
Since we know that the sum of the angles in a triangle is 180 degrees, we can find angle I.
In this case, angle I is 180 - 90 - G (angle H is a right angle).
angle I = 180 - 90 - G
angle I = 90 - G
angle I = 90 - cos^-1(40/41) (using the inverse cosine function)
To find sin(I), we can use the sine ratio:
sin(I) = HI / HI
sin(I) = HI / HI
Since HI is the hypotenuse, sin(I) = 1.
So the value of sin(I) is 1.
c) Cot G:
The cotangent is defined as cot(angle) = adjacent side / opposite side.
In this case, we have G as angle G and GH as the adjacent side and HI as the opposite side.
cot(G) = GH / HI
cot(G) = 40 / 41
So the value of cot(G) is 40/41 or approximately 0.9756.
d) csc G:
The cosecant is defined as csc(angle) = hypotenuse / opposite side.
In this case, we have G as angle G, HI as the hypotenuse, and GH as the opposite side.
csc(G) = HI / GH
csc(G) = 41 / 40
So the value of csc(G) is 41/40 or approximately 1.025.
e) cos I:
To find cos(I), we can use the cosine ratio:
cos(I) = GH / HI
cos(I) = 40 / 41
So the value of cos(I) is 40/41 or approximately 0.9756.
f) sec H:
The secant is defined as sec(angle) = hypotenuse / adjacent side.
In this case, we have H as angle H, HI as the hypotenuse, and GH as the adjacent side.
sec(H) = HI / GH
sec(H) = 41 / 40
So the value of sec(H) is 41/40 or approximately 1.025.
I hope this explanation helps you understand how to solve this problem! Let me know if you have any further questions.