A. 0.72
Step-by-step explanation:
Short course in trigonometry
Trig functions (sin, cos, tan) relate an angle to the ratios of particular side lengths in a right triangle. We often use the mnemonic SOH CAH TOA to help remember the relationships ...
Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
The sides are considered "opposite" or "adjacent" with respect to a given acute angle in the right triangle. (See the attachment.)
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Special Triangles
There are a couple of "special triangles" that have trig function values that are relatively easy to remember. One of these is the isosceles (45°-45°-90°) right triangle, whose sides are in the ratios 1 : 1 : √2. Another is the 30°-60°-90° right triangle, whose sides are in the ratios 1 : √3 : 2. Knowing that the smallest angle is always opposite the shortest side, and that the hypotenuse is always the longest side, we can use these relations to find the trig functions of 30°, 45°, and 60° angles.
For example, ...
sin(45°) = opposite/hypotenuse = 1/√2
tan(30°) = opposite/adjacent = 1/√3
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Inverse trig functions
When you have a trig ratio (say sin(x) = 0.8) and you want to find the corresponding angle, you need to make use of the inverse sine function. This is often written as sin⁻¹ and it can go by the name arcsine. When you read an expression like ...
x = sin⁻¹(0.8)
you can read it as "x is the angle whose sine is 0.8". Scientific and graphing calculators have function keys for these functions as well as for the trig functions sin, cos, tan. Often a "shift" or "2nd" key is required to access these inverse functions. There is usually a mode setting that tells the calculator whether to give the angle in degrees or radians.
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Solving triangles
When you have a right triangle, the above relationships are generally all you need to find the values of all sides and angles (solve the triangle). The Pythagorean theorem can be helpful, too.
When you do not have a right triangle (as in this problem), there are other relationships that are used to solve the triangle. One of these is the Law of Sines; another is the Law of Cosines.
The Law of Cosines is useful when you know all the side lengths and you don't know any of the angles (as here).
For a triangle with sides a, b, c and opposite angles A, B, C, the relationship is given by the formula ...
a² = b² +c² -2bc·cos(A)
You will notice when you draw this triangle that angle A is between sides b and c. The law is completely symmetrical, so any sides can be named any letters. Using the above equation in this problem, it is convenient to assign ...
a = 7, b = 8, c = 10
We do this because side 7 is opposite the angle θ, just as side "a" is opposite angle A in the formula. Using the formula with these variable assignments gives us a way to find cos(θ), which is what the question is asking for.
If you like, you can solve the above equation for cos(θ) before doing anything with numbers:
cos(θ) = (b² +c² -a²)/(2bc)
Filling in the numbers, we get ...
cos(θ) = (8² +10² -7²)/(2·8·10) = 115/160 = 0.71875 ≈ 0.72
cos(θ) = 0.72
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Additional comment
If you were to use the arccos function to find the angle, you would learn that it is about 44.049°. A scale drawing of this triangle is shown in the second attachment.
