We are told that in △ GHI, ∠H is a right angle and the length of the segment GH is 40.
We are also given the following trigonometric expression.
cos G =40/41Recall that in a right triangle, the cosine of an acute angle is defined as the ratio of the adjacent side to the hypotenuse.
cos θ = Adjacent/Hypotenuse
From the diagram, we know that the length of the adjacent side to G is 40. Then, if we compare the above ratio with the given expression we can see that the hypotenuse has to be 41.
cos θ = Adjacent/Hypotenuse ⇒ cos G = 40/41
Now, we can find the length of the opposite side by substituting a= 40 and c= 41 into the Pythagorean Theorem.
Next, we will draw the right triangle and label its three sides.
We want to find the value of sin G. To do so, we will recall the trigonometric ratio for sine.
sin θ = Opposite/Hypotenuse
We know that the length of the opposite side is 9 and the length of the hypotenuse is 41.
sin θ = Opposite/Hypotenuse ⇒ sin G = 9/41
Let's find the value in decimal form.
We want to find the value of sin I. To do so, we will recall the trigonometric ratio for sine.
sin θ = Opposite/Hypotenuse
In this case, the length of the opposite side is 40 and the length of the hypotenuse is 41.
We can rewrite the trigonometric ratio by substituting these values.
sin θ = Opposite/Hypotenuse ⇒ sin I = 40/41
Now, we will find the value in decimal form.
We want to find the value of cot G. To do it, we will recall the trigonometric ratio for cotangent.
cot θ = Adjacent/Opposite
For this case, the length of the adjacent side is 40, the length of the opposite side is 9 and the angle is G.
Now, we will rewrite the trigonometric ratio for cotangent by substituting these values.
cot θ = Adjacent/Opposite
⇒ cot G = 40/9
The next step will be finding the value in decimal form.
Now, we want to find the value of csc G. To do so, let's recall the trigonometric ratio for cosecant.
csc θ = Hypotenuse/Opposite
We know that the length of the opposite side is 9, the length of the hypotenuse is 41, and the angle is G.
The next step will be rewriting the trigonometric ratio for cosecant by substituting these values.
csc θ = Hypotenuse/Opposite
⇒ csc G = 41/9
Let's calculate the value in decimal form.
Let's find the value of cos I by recalling the trigonometric ratio for cosine.
cos θ = Adjacent/Hypotenuse
In this case, the length of the hypotenuse is 41, the length of the adjacent side is 9, and the angle is I.
Next, we will rewrite the trigonometric ratio by substituting these values.
cos θ = Adjacent/Hypotenuse
⇒ cos I = 9/41
We will find this value in decimal form. Let's do it!
Notice that the ∠H is a right angle, which means that is equal to 90^(∘). We cannot calculate the value of sec 90 ^(∘) because it will be undefined. This means that is not possible to represent it with a number. To show it, we will recall the reciprocal identity for secant.
sec θ = 1/cos θ
Now, we will calculate this equation by substitute θ= 90^(∘).
