What Are the Solutions to the Equation (sin 2x+cos 2x)²=1?
Answer – The solutions to the equation (sin 2x + cos 2x)² = 1 are x = (nℼ)/4 with x = 0, ℼ/2, ℼ, (3ℼ)/2, 2ℼ… for even integers (n = 0, 2, 4, 6, 8…) and x = ℼ/4, (3ℼ)/4, (5ℼ)/4, (7ℼ)/4, (9ℼ)/4… for odd integers (n = 1, 3, 5, 7, 9…).
Explanation:
Let’s follow a step-by-step approach to solve the given equation.
First, we expand the equation:
(sin 2x + cos 2x)(sin 2x + cos 2x) = 1
Next, we multiply the terms on the left side:
sin ²(2x) + sin 2x cos 2x + cos 2x sin 2x + cos²(2x) = 1
We add the common terms:
sin ²(2x) + 2 sin 2x cos 2x + cos²(2x) = 1
Then, we rearrange the obtained equation:
sin ²(2x) + cos²(2x) + 2 sin 2x cos 2x = 1
Now, using the trigonometric identity sin²θ + cos²θ = 1 in the above equation, we get:
1 + 2 sin 2x cos 2x = 1
On simplifying further:
2 sin 2x cos 2x = 0
sin 2x cos 2x = 0
Therefore:
sin 2x = 0 – Equation 1
cos 2x = 0 – Equation 2
Also, from the trigonometric values of special angles:
sin 0 = 0 – Equation 3
cos ℼ/2 = 0 – Equation 4
From Equations 1 and 3, we get:
2x = 0
x = 0
Further, from Equations 2 and 4, we get:
2x = ℼ/2
x = ℼ/4
Hence, x = (nℼ)/4, where n = 0, 1, 2, 3, 4…
Thus, for even integers (n = 0, 2, 4, 6, 8…), the solutions to the given equation are x = 0,ℼ/2,ℼ,(3ℼ)/2,2ℼ…
And for odd integers (n = 1, 3, 5, 7, 9…), the solutions to the given equation are x = ℼ/4,3ℼ/4,5ℼ/4,7ℼ/4,9ℼ/4…
Popular Questions
- A Diginacci sequence is created as follows. The first two terms are any positive whole numbers. Each of the remaining terms is the sum of the digits of the previous two terms For example, starting with 4 and 8 the Diginacci sequence is 4, 8, 13, 12, 7, 10, … The calculations for this exmaple are 4 + 8 = 13, 8 + 1 + 3 = 12, 1 + 3 + 1 + 2 = 7, 1 + 2 + 7 = 10 a. Calculate the first 28 terms of the Diginacci sequence with starting terms 1 and 1, and then find the 2021st number in the sequence. b. Show that if both starting terms in a Diginacci sequence are ach less than one million, then its fourth and fifth terms are each less than 100. c. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms are at most 20. d. Show that if both starting terms in a Diginacci sequence are each less than 100, then it has a term after which all terms equal 18 or all terms are less than 18.
SHOW MORE TEXTBOOK SOLUTIONS+
-
1. Solve x2 = 2.
- Determine the type of number that the solutions of x2+2x+5=0 will be.
- Solve each equation in Exercises 96-102 by the method of your choice.
97.
- In Problems 11 68, solve each equation. x2=9x
- For Exercises 26–36, solve the equations. (See Examples 4–5.) x 2 x − 2 = 1 x − 2
- Solve each absolute value equation
25.
- Solve the equations by using the square root property. Express any complex numbers using i notation.
5.
- Solve.
15.
- For Exercises 26–36, solve the equations. (See Examples 4–5.)
28.
- Find the real solutions, if any, of the equation
- Solve. x22x=2
-
3. Solve x2 – 4x = 1 by completing the square.
- For Exercises 5 16, solve the equation. (See Example 1) 2x=32
- Solve x2 – 2x = 48 by factoring.
- Compute 4cos70+isin703 and write the result in rectangular form a + bi.
- In Exercises 91-114, solve each quadratic equation by the method of your choice. (2x5)(x+1)=2
- Solve each equation in x for exact solutions over the interval [0, 2) and each equation in for exact solutions over the interval [0, 360). See Exa...
- For Exercises 2–7, solve the quadratic equations. 2 x 2 − 8 x − 44 = 0
- Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimensi...
- For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 24cos6+isin6
- Solve the equation 1+2cos2=0 . a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 .
- Use the double-angle formula cos2x=2cos2x1 to evaluate the integral 11+cos2dx
- In Problems 8186, solve each equation by the Square Root Method. (x+2)2=1
- Problems are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you...
- Solve. 2 x ( x − 7 ) = x − 18
- A. Solve. 2x + 6 = 0
- Find the following. (2y1)2dy
-
Solve each equation for exact solutions. See Example 4.
39.
- Solve the equations and check your solutions. If there is no solution, say so
1.
- In Exercises 31–50, solve each equation by the method of your choice. Simplify solutions, if possible.
- In problems 16, solve each equation. 2x2x=0
- Find the real solutions, if any, of the equation 2x2+x1=0
- Solve.
7. [5.7]
- Use the method of your choice to solve each equation. See the strategy for solving ax2 + bx + c = 0 on page 138. x 2 = 4 3 x + 5 9
- Solve.
125.
- For exercises 97-100, use your knowledge of the graphs of the sine function and linear functions to determine the number of solutions to the equati...
- 10. Solve by completing the square. Show your work.
- Solve. (Find all complex-number solutions.) u2+2u4=0
- Write each complex number in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°). See Example 3.
49. 2 + 2i
-
Solve each equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. See Exam...
- Solve. [15.1c]
16.
- (a) Evaluate the integral sinxcosxdx by two methods: first by letting u=sinx , and then by letting u=cosx . (b) Explain why the two apparently diff...
- Solve Problems 1-4 by the square-root method. 2x222=0
- Solve each exponential equation in Exercises 1-18 by expressing each side as a power of the same base and then equating exponents.
- Solve, finding all solutions in [0, 2π).
2 sec x tan x + 2 sec x + tan x + 1 = 0
- a Solve. Don’t forget to check!
21.