Find the remaining trigonometric value ∅ if cos∅= 16/20
and ∅ is an angle in a right triangle
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Find the remaining trigonometric value ∅ if cos∅= 16/20
and ∅ is an angle in a right triangle
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Given that cos ∅ = 16/20 and ∅ is an angle in a right triangle, we can use the Pythagorean theorem and the definition of cosine to find the remaining trigonometric value.
Let's start by drawing a right triangle and labeling its sides as follows:
Adjacent side = 16 (since cosine is adjacent over hypotenuse)
Hypotenuse = 20 (given)
Opposite side = ?
Using the Pythagorean theorem, we can find the length of the opposite side:
Opposite side^2 = Hypotenuse^2 - Adjacent side^2
Opposite side^2 = 20^2 - 16^2
Opposite side^2 = 144
Opposite side = √144
Opposite side = 12
Now we can use the definition of sine to find the value of sin ∅:
sin ∅ = Opposite side / Hypotenuse
sin ∅ = 12 / 20
sin ∅ = 3/5
Therefore, the remaining trigonometric value is sin ∅ = 3/5.
Step-by-step explanation:
We know that cos∅ = adjacent/hypotenuse in a right triangle.
So, let's assume that the adjacent side in our triangle is 16 and the hypotenuse is 20.
Using the Pythagorean Theorem, we can find the opposite side:
opposite^2 = hypotenuse^2 - adjacent^2
opposite^2 = 20^2 - 16^2
opposite^2 = 144
opposite = √144
opposite = 12
Now we have all three sides of the triangle and can use any trigonometric function to find the remaining angles.
Using the sine function, we get:
sin∅ = opposite/hypotenuse
sin∅ = 12/20
sin∅ = 0.6
Taking the inverse sine (or arcsine) of 0.6, we get:
∅ = sin⁻¹(0.6)
∅ = 36.87 degrees (rounded to two decimal places)
Therefore, the remaining trigonometric value ∅ is 36.87 degrees.