Unit Circle Problems solve cosW=sin20, sinW=cos(-10), sinW< 0.5 and 1<tanW

• MHB
• Naidely
In summary, to find all solutions w between 0 and 360, inclusive, we can first use the unit circle to determine that cosW=sin20 for an angle of 70 degrees. Then, for sinW=cos(-10), we can use the graphs of y=sin(x) and y=cos(x) to find solutions at 80 and 260 degrees. Finally, by comparing the graphs of y=sin(x) and y=1/2, and y=tan(x) and y=1, we can determine that sinW<0.5 for angles of 30 and 150 degrees, and 1<tanW for angles of 45 and 225 degrees.
Naidely
Without a calculator, find all solutions w between 0 and 360, inclusive, providing diagrams that support your results.

1) cosW=sin20

2) sinW=cos(-10)

3) sinW< 0.5

4) 1<tanW

You titled this "unit circle problems". Have you drawn a unit circle? Do you know that, at each (x, y) point on the circle, $$x= cos(\theta)$$ and $$y= sin(\theta)$$? Where is sin(20) on that circle? So for what angle is cos(W)= sin(20)?

yes I am aware that y= sin(theta) and x= cos(theta)
and yes I drew my unit circle
so sin20 is in the first quadrant but I am not sure how to import images here, if possible
and cos70= sin 20, I figured that out a few days ago, however I am stuck on 3 and 4

Instead of using a unit circle, for 3 and 4, draw the graphs of y= sin(x) and y= tan(x). Compare the graph of y= sin(x) with the graph of y= 1/2 and compare the graph of y= tan(x) with y= 1. You should know that sin(x)= 1/2 for $$\pi/6$$ radians (30 degrees) and $$5\pi/6$$ radians (150 degrees) and that tan(x)= 1 for $$\pi/4$$ radians (45 degrees) and $$5\pi/4$$ radians (225 degrees). (There is a nice graphing app at https://www.desmos.com/calculator.)

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 on a Cartesian plane. It is used in trigonometry to represent the relationship between the angles and sides of a right triangle. The coordinates of points on the unit circle correspond to the sine and cosine values of the angles in the triangle.

How do I solve for an unknown angle using the unit circle?

To solve for an unknown angle using the unit circle, you can use the inverse trigonometric functions (sin-1, cos-1, tan-1). Plug in the given values for the sine, cosine, or tangent of the angle and use the inverse function to find the measure of the angle.

What does it mean if sinW is less than 0.5 in a unit circle problem?

If sinW is less than 0.5, it means that the angle W is less than 30 degrees or π/6 radians. This is because the sine function of an angle measures the ratio of the opposite side to the hypotenuse in a right triangle, and when this ratio is less than 0.5, the angle must be less than 30 degrees.

What is the relationship between sine and cosine values on the unit circle?

The sine and cosine values on the unit circle are related by the Pythagorean identity, sin2θ + cos2θ = 1. This means that the square of the sine value plus the square of the cosine value will always equal 1 on the unit circle.

How do I solve for an angle that satisfies multiple trigonometric equations on the unit circle?

To solve for an angle that satisfies multiple trigonometric equations, you can use the same method as solving a single equation. Use the given values for the sine, cosine, and/or tangent of the angle to find its measure on the unit circle. If the angle satisfies all the given equations, it will be the solution to the problem.

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