Trigonometry Problem: Finding sin2Ө Given sinӨ + cosӨ = 4/3

In summary: Thanks!In summary, the student is trying to solve for cosθ and sinθ but realizes that it won't be unique. He tries a method that involves squaring both sides of the equation.
  • #1
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Homework Statement


sinӨ + cosӨ =4/3 ,, then sin2Ө = ... ?

Homework Equations


sin2Ө = 2cosӨsinӨ
or any trigonometry functions...

The Attempt at a Solution


I tried to write sinӨ + cosӨ =4/3 in the form of cosӨsinӨ but couldn't..
is there a possible way to solve it ?
 
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  • #2
sin2θ+cos2θ=1
 
  • #3
Nathanael said:
sin2θ+cos2θ=1
how is that possible ?
 
  • #4
I'm sorry for the unhelpful post, I was assuming you knew that equation and were just forgetting it.

The equation comes from Pythagorean's theorem and it's always true for all θ (so you can use it in

Imagine (or draw) the Unit Circle. The x coordinate of a point on the circle is cosθ, and the y coordinate is sinθ (where θ is the angle between the point and the positive x-axis). We know from Pythagorean's theorem that x2+y2=1 (because 1 is the radius of the circle) therefore sin2θ+cos2θ=1

Edit:
Or I could have said, the equation of the unit circle is x2+y2=1, therefore sin2θ+cos2θ=1
 
  • #5
Nathanael said:
I'm sorry for the unhelpful post, I was assuming you knew that equation and were just forgetting it.

The equation comes from Pythagorean's theorem and it's always true for all θ (so you can use it in

Imagine (or draw) the Unit Circle. The x coordinate of a point on the circle is cosθ, and the y coordinate is sinθ (where θ is the angle between the point and the positive x-axis). We know from Pythagorean's theorem that x2+y2=1 (because 1 is the radius of the circle) therefore sin2θ+cos2θ=1

Edit:
Or I could have said, the equation of the unit circle is x2+y2=1, therefore sin2θ+cos2θ=1
yeah i know this equation , what i meant is how can i use it to solve this problem ?
i tried it but didnt get the answer :(
 
  • #6
sin2θ+cos2θ=1
sinθ+cosθ=4/3

I haven't done the manipulations, but that is 2 equations with 2 unknowns; it should be enough to uniquely determine cosθ and sinθ
(In other words, there should be only 1 pair of cosθ and sinθ which add to 4/3 while also having their squares add to one.)
 
  • #7
I just realized, it won't uniquely determine cosθ and sinθ, because you could simply interchange cosθ and sinθ and you will have another solution. But this shouldn't change the answer since your formula for sin(2θ) involves the product of the two (which remains unchanged if you interchange them).
 
  • #8
Nathanael said:
sin2θ+cos2θ=1
sinθ+cosθ=4/3

I haven't done the manipulations, but that is 2 equations with 2 unknowns; it should be enough to uniquely determine cosθ and sinθ
(In other words, there should be only 1 pair of cosθ and sinθ which add to 4/3 while also having their squares add to one.)
oh i got it know ,, thanks a lot
 
  • #9
No problem
 
  • #10
I assumed you were suggesting he square both sides of his equation: if [itex]sin(x)+ cos(x)= 4/3[/itex] so [itex](sin(x)+ cos(x))^2= sin^2(x)+ 2sin(x)cos(x)+ cos^2(x)= 16/9[/itex]. But [itex]sin^2(x)+ cos^2(x)= 1[/itex] so this is [itex]2sin(x)cos(x)+ 1= 16/9[/itex], [itex]2sin(x)cos(x)= 7/9[/itex].
 
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  • #11
HallsofIvy said:
I assumed you were suggesting he square both sides of his equation: if [itex]sin(x)+ cos(x)= 4/3[/itex] so [itex](sin(x)+ cos(x))^2= sin^2(x)+ 2sin(x)cos(x)+ cos^2(x)= 16/9[/itex]. But [itex]sin^2(x)+ cos^2(x)= 1[/itex] so this is [itex]2sin(x)cos(x)+ 1= 16/9[/itex], [itex]2sin(x)cos(x)= 7/9[/itex].
Nice method :)
 

1. What is trigonometry?

Trigonometry is a branch of mathematics that focuses on the study of triangles and the relationships between their sides and angles. It involves using ratios and functions to solve problems involving triangles in a variety of contexts.

2. What are the three basic trigonometric functions?

The three basic trigonometric functions are sine, cosine, and tangent. These functions are used to relate the angles of a triangle to the lengths of its sides. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

3. How do I find the missing side length in a right triangle using trigonometry?

To find a missing side length in a right triangle, you can use the Pythagorean theorem or one of the trigonometric functions. If you know the lengths of two sides and are looking for the third, you can use sine, cosine, or tangent depending on which sides and angle are given. If you know two angles and one side, you can use the sine or cosine functions to find the missing side length.

4. How is trigonometry used in real life?

Trigonometry has many practical applications in fields such as engineering, physics, and navigation. It is used to measure and analyze angles and distances, as well as to design and construct buildings, bridges, and other structures. Trigonometry is also used in astronomy to calculate the positions and movements of celestial bodies.

5. What is the unit circle and how is it used in trigonometry?

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used in trigonometry to define the trigonometric functions for any angle, not just those in a right triangle. The x-coordinate on the unit circle represents the cosine of an angle, and the y-coordinate represents the sine. The unit circle is also used to find the values of trigonometric functions for special angles, such as 30°, 45°, and 60°.

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