can I please have help with these 3 questions?
1. Solve this equation graphically on the interval [0, 2ð]. list the solutions.
sin(2x)-1=tan x
2. solve sin x cos x= sqrt3/4
3. solve tan^2 x-3tan x+2=0
thank you! show step by step please.
1. try
http://www.wolframalpha.com/input/?i=sin%282x%29-1%3Dtan%28x%29+
2.
sinxcosx = √3/4
times 2
2sinxcosx = √3/2
sin(2x) = √3/2
2x = 60° or 2x = 120°
x = 30° or x = 60°
period of sin 2x is 180°
so other solutions
x = 30 + k(180) and x = 60 + k(180) degrees , where k is an inter
in radians:
x = π/6 + kπ or x = π/3 + kπ
3. let tanx = t
you have t^2 - 3t + 2 = 0
(t-2)(t-1) = 0
t = 2 or t = -1
so
tanx = 2 -----> two answers in domain 0≤x≤2π
or
tanx = -1 ---> two answers in domain 0 ≤ x ≤ 2
give it a try
thank you so much!
Of course! I'll be glad to help you solve these three questions step by step.
1. Solve the equation graphically: sin(2x) - 1 = tan(x)
To solve this equation graphically, you can plot the graphs of y = sin(2x) - 1 and y = tan(x) on the same coordinate plane and find the points where they intersect.
1. First, graph y = sin(2x) - 1:
- Take your interval [0, 2π] and mark several points. For instance, you can mark 0, π/2, π, 3π/2, and 2π on the x-axis.
- Substitute these values of x into the equation y = sin(2x) - 1 to find the corresponding y-values and plot the points.
2. Next, graph y = tan(x):
- Again, mark the same points on the x-axis as in step 1.
- Plug in the x-values into the equation y = tan(x) to get the corresponding y-values. Plot these points on the same graph.
3. Finally, identify the points where the two curves intersect. These points represent the solutions to the equation.
2. Solve the equation: sin(x)cos(x) = √3/4
To solve this equation analytically, follow these steps:
1. Recognize that √3/4 is equivalent to tan(π/3). Therefore, we can rewrite the equation as sin(x)cos(x) = tan(π/3).
2. Convert sin(x)cos(x) to its equivalent form using double-angle identities: (sin(2x))/2 = tan(π/3).
3. Apply the inverse tangent to both sides of the equation to isolate x:
- arctan((sin(2x))/2) = arctan(tan(π/3)).
4. Use the identity tan(arctan(t)) = t to simplify the right side:
- arctan((sin(2x))/2) = π/3.
5. Solve for x by taking the inverse tangent of both sides:
- (sin(2x))/2 = tan(π/3).
- sin(2x) = 2tan(π/3).
- sin(2x) = 2tan(60°).
- sin(2x) = 2(√3).
6. Solve for x by finding the arcsine of both sides:
- 2x = arcsin(2√3).
- x = (1/2)arcsin(2√3) + πn, with n being an integer.
3. Solve the equation: tan^2(x) - 3tan(x) + 2 = 0
To solve this quadratic equation, follow these steps:
1. Factor the equation (if possible) by finding two binomials that multiply together to give you the quadratic equation. In this case, we have (tan(x) - 2)(tan(x) - 1) = 0.
2. Set each factor equal to zero and solve:
- tan(x) - 2 = 0, giving us tan(x) = 2.
- tan(x) - 1 = 0, giving us tan(x) = 1.
3. Find the solutions for x by taking the inverse tangent of both sides:
- x = arctan(2) + πn, with n being an integer.
- x = arctan(1) + πn, with n being an integer.
Please let me know if you have any further questions!