In triangle GHI, angle H is a right angle, GH=40, and cos G=40/41. 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
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To find the values in fraction and decimal form, we'll use the following trigonometric identities:
1. Pythagorean Identity: sin²θ + cos²θ = 1
2. Sine and Cosine Ratios: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse
3. Tangent Ratio: tanθ = opposite/adjacent, cotθ = adjacent/opposite
4. Cosecant Ratio: cscθ = 1/sinθ, secθ = 1/cosθ
Let's start with the given information:
In triangle GHI, angle H is a right angle (90 degrees), GH = 40, and cos G = 40/41.
a. To find sin G:
Using the Pythagorean Identity, sin²G + cos²G = 1
sin²G + (40/41)² = 1
sin²G = 1 - (40/41)²
sinG = √[1 - (40/41)²]
sinG ≈ √[1 - 1600/1681]
sinG ≈ √[81/1681]
sinG ≈ 9/41 ≈ 0.2195 (approx.)
b. To find sin I:
Since angle H is a right angle, angle I is 180 - 90 - G = 90 - G.
Therefore, sin I = sin(90 - G) = cosG (due to the complementary property of sine and cosine)
sin I = cos G = 40/41
c. To find cot G:
cot G = adjacent/opposite = GH/IG
cot G = 40/IG
d. To find csc G:
csc G = 1/sin G = 1/(9/41) = 41/9
e. To find cos I:
cos I = adjacent/hypotenuse = IG/HI
f. To find sec H:
sec H = 1/cos H = 1/cos 90 = 1/0 (undefined)
Please note that we can't find the values of cos I and sec H without additional information about angle I and HI.
254
if cosG = 40/41, sinG = 9/41
and the rest follow easily