- #1
Vee9
- 27
- 0
Homework Statement
Solve for θ.
2sinθcosθ + 1 - 2sin^2θ = 0
The Attempt at a Solution
I replaced 2sinxcoxx with sin2x.
so I ended up with:
-2sin^2(x) + sin2x + 1 = 0
How should I continue to solve for theta? (or x)
Bohrok said:You also could write it as [itex]\cos 2x + \sin 2x = 0 \Longrightarrow sin2x = -cos2x \Longrightarrow tan2x = -1[/itex]
Vee9 said:The answer is 67.5.
So from the tan2x = -1 that you found, I did:
45+ (45/2) = 67.5
One thing I didn't understand was how you got
tan2x = -1 from sin2x = -cos2x ?
Thanks. Major mind block today, lol.
jambaugh said:There are multiple paths.
Apply both half/double angle identities, i.e. also [tex]\sin^2(\theta)=(1-\cos(2\theta))/2[/tex]
This gives you an equation you can solve directly because of its simple form.
One other approach is to not use half angle identities but to use the base pythagorean identity on the cosine. You'll get a square root so solve for that term and square it out. The result will be a quartic equation but its really a quadratic equation in sine squared of theta so you can solve it easily.
To identify the factors of a number, you can start by listing all the possible pairs of numbers that multiply to give the given number. For example, if the number is 12, you can list (1,12), (2,6), (3,4). These pairs of numbers are the factors of 12.
A prime number is a number that is divisible by only 1 and itself, while a composite number is a number that has more than two factors. In simpler terms, prime numbers are numbers that can only be divided evenly by 1 and itself, while composite numbers have multiple factors.
To factor a polynomial expression, you can use techniques such as factoring by grouping, factoring using the difference of squares, or factoring using the sum or difference of cubes. It involves breaking down the polynomial into smaller parts that can be multiplied together to get the original polynomial.
Yes, you can use a calculator to factor a number or expression. Many scientific calculators have a built-in function for factoring. However, it is important to note that using a calculator may not always give you the most simplified form of the factors, so it is always a good idea to double-check your answer using other methods.
Factoring is an important concept in mathematics that helps us simplify and solve equations, find the roots of polynomials, and identify common factors in algebraic expressions. It is also used in many real-world applications, such as finding the least common multiple in fractions and simplifying complex formulas in physics and engineering.