Trigonometry Examples

Find the Angle Between the Vectors (1,5/3) , (1,-8)
,
Step 1
Use the dot product formula to find the angle between two vectors.
Step 2
Find the dot product.
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Step 2.1
The dot product of two vectors is the sum of the products of the their components.
Step 2.2
Simplify.
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Step 2.2.1
Simplify each term.
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Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Multiply .
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Step 2.2.1.2.1
Combine and .
Step 2.2.1.2.2
Multiply by .
Step 2.2.1.3
Move the negative in front of the fraction.
Step 2.2.2
Write as a fraction with a common denominator.
Step 2.2.3
Combine the numerators over the common denominator.
Step 2.2.4
Subtract from .
Step 2.2.5
Move the negative in front of the fraction.
Step 3
Find the magnitude of .
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Step 3.1
The norm is the square root of the sum of squares of each element in the vector.
Step 3.2
Simplify.
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Step 3.2.1
One to any power is one.
Step 3.2.2
Apply the product rule to .
Step 3.2.3
Raise to the power of .
Step 3.2.4
Raise to the power of .
Step 3.2.5
Write as a fraction with a common denominator.
Step 3.2.6
Combine the numerators over the common denominator.
Step 3.2.7
Add and .
Step 3.2.8
Rewrite as .
Step 3.2.9
Simplify the denominator.
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Step 3.2.9.1
Rewrite as .
Step 3.2.9.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4
Find the magnitude of .
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Step 4.1
The norm is the square root of the sum of squares of each element in the vector.
Step 4.2
Simplify.
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Step 4.2.1
One to any power is one.
Step 4.2.2
Raise to the power of .
Step 4.2.3
Add and .
Step 5
Substitute the values into the formula.
Step 6
Simplify.
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Step 6.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.2
Combine and .
Step 6.3
Simplify the numerator.
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Step 6.3.1
Combine using the product rule for radicals.
Step 6.3.2
Multiply by .
Step 6.4
Multiply the numerator by the reciprocal of the denominator.
Step 6.5
Multiply by .
Step 6.6
Cancel the common factor of .
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Step 6.6.1
Move the leading negative in into the numerator.
Step 6.6.2
Cancel the common factor.
Step 6.6.3
Rewrite the expression.
Step 6.7
Combine and .
Step 6.8
Move the negative in front of the fraction.
Step 6.9
Multiply by .
Step 6.10
Combine and simplify the denominator.
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Step 6.10.1
Multiply by .
Step 6.10.2
Raise to the power of .
Step 6.10.3
Raise to the power of .
Step 6.10.4
Use the power rule to combine exponents.
Step 6.10.5
Add and .
Step 6.10.6
Rewrite as .
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Step 6.10.6.1
Use to rewrite as .
Step 6.10.6.2
Apply the power rule and multiply exponents, .
Step 6.10.6.3
Combine and .
Step 6.10.6.4
Cancel the common factor of .
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Step 6.10.6.4.1
Cancel the common factor.
Step 6.10.6.4.2
Rewrite the expression.
Step 6.10.6.5
Evaluate the exponent.
Step 6.11
Evaluate .