If { \sin 31^ \circ} = P$}$, Determine The Following, Without Using A Calculator 1. { \sin 149^{\circ $}$2. { \cos(-59^{\circ})$}$3. { \cos 62^{\circ}$}$Simplify The Following Expression To A Single Trigonometric

by ADMIN 213 views

If sin⁡31∘=p\sin 31^{\circ} = p, Determine the Following Trigonometric Values Without Using a Calculator

In this article, we will explore the world of trigonometry and determine the values of various trigonometric functions without using a calculator. We will start by using the given value of sin⁡31∘=p\sin 31^{\circ} = p and then use trigonometric identities to find the values of sin⁡149∘\sin 149^{\circ}, cos⁡(−59∘)\cos(-59^{\circ}), and cos⁡62∘\cos 62^{\circ}. We will also simplify an expression to a single trigonometric function.

Before we begin, let's recall some important trigonometric identities that we will use in this article:

  • sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x
  • cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x
  • sin⁡(360∘−x)=sin⁡x\sin (360^{\circ} - x) = \sin x
  • cos⁡(360∘−x)=cos⁡x\cos (360^{\circ} - x) = \cos x
  • sin⁡(90∘+x)=cos⁡x\sin (90^{\circ} + x) = \cos x
  • cos⁡(90∘+x)=−sin⁡x\cos (90^{\circ} + x) = -\sin x

Determine the Value of sin⁡149∘\sin 149^{\circ}

We are given that sin⁡31∘=p\sin 31^{\circ} = p. We can use the identity sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x to find the value of sin⁡149∘\sin 149^{\circ}.

sin⁡149∘=sin⁡(180∘−31∘)\sin 149^{\circ} = \sin (180^{\circ} - 31^{\circ}) =sin⁡31∘= \sin 31^{\circ} =p= p

Therefore, the value of sin⁡149∘\sin 149^{\circ} is pp.

Determine the Value of cos⁡(−59∘)\cos(-59^{\circ})

We can use the identity cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x to find the value of cos⁡(−59∘)\cos(-59^{\circ}).

cos⁡(−59∘)=cos⁡(180∘−121∘)\cos(-59^{\circ}) = \cos (180^{\circ} - 121^{\circ}) =−cos⁡121∘= -\cos 121^{\circ} =−cos⁡(180∘−59∘)= -\cos (180^{\circ} - 59^{\circ}) =−(−cos⁡59∘)= -(-\cos 59^{\circ}) =cos⁡59∘= \cos 59^{\circ}

We can use the identity cos⁡(90∘+x)=−sin⁡x\cos (90^{\circ} + x) = -\sin x to find the value of cos⁡59∘\cos 59^{\circ}.

cos⁡59∘=cos⁡(90∘+31∘)\cos 59^{\circ} = \cos (90^{\circ} + 31^{\circ}) =−sin⁡31∘= -\sin 31^{\circ} =−p= -p

Therefore, the value of cos⁡(−59∘)\cos(-59^{\circ}) is −p-p.

Determine the Value of cos⁡62∘\cos 62^{\circ}

We can use the identity cos⁡(90∘+x)=−sin⁡x\cos (90^{\circ} + x) = -\sin x to find the value of cos⁡62∘\cos 62^{\circ}.

cos⁡62∘=cos⁡(90∘+28∘)\cos 62^{\circ} = \cos (90^{\circ} + 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−62∘)= -\sin (90^{\circ} - 62^{\circ}) =−sin⁡62∘= -\sin 62^{\circ} =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} =−sin⁡(90∘−(90∘−62∘))= -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =−sin⁡(90∘−28∘)= -\sin (90^{\circ} - 28^{\circ}) =−sin⁡28∘= -\sin 28^{\circ} $= -\sin (90^\circ}
**Q&A Trigonometry and the Value of $\sin 31^{\circ = p$**

In our previous article, we explored the world of trigonometry and determined the values of various trigonometric functions without using a calculator. We started by using the given value of sin⁡31∘=p\sin 31^{\circ} = p and then used trigonometric identities to find the values of sin⁡149∘\sin 149^{\circ}, cos⁡(−59∘)\cos(-59^{\circ}), and cos⁡62∘\cos 62^{\circ}. In this article, we will answer some frequently asked questions about trigonometry and the value of sin⁡31∘=p\sin 31^{\circ} = p.

Q: What is the value of sin⁡31∘\sin 31^{\circ}?

A: The value of sin⁡31∘\sin 31^{\circ} is given as pp.

Q: How did you determine the value of sin⁡149∘\sin 149^{\circ}?

A: We used the identity sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x to find the value of sin⁡149∘\sin 149^{\circ}. Since sin⁡31∘=p\sin 31^{\circ} = p, we have sin⁡149∘=sin⁡(180∘−31∘)=sin⁡31∘=p\sin 149^{\circ} = \sin (180^{\circ} - 31^{\circ}) = \sin 31^{\circ} = p.

Q: How did you determine the value of cos⁡(−59∘)\cos(-59^{\circ})?

A: We used the identity cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x to find the value of cos⁡(−59∘)\cos(-59^{\circ}). Since cos⁡59∘=−sin⁡31∘=−p\cos 59^{\circ} = -\sin 31^{\circ} = -p, we have cos⁡(−59∘)=cos⁡59∘=−p\cos(-59^{\circ}) = \cos 59^{\circ} = -p.

Q: How did you determine the value of cos⁡62∘\cos 62^{\circ}?

A: We used the identity cos⁡(90∘+x)=−sin⁡x\cos (90^{\circ} + x) = -\sin x to find the value of cos⁡62∘\cos 62^{\circ}. Since sin⁡28∘=sin⁡(90∘−62∘)=sin⁡62∘\sin 28^{\circ} = \sin (90^{\circ} - 62^{\circ}) = \sin 62^{\circ}, we have $\cos 62^{\circ} = -\sin 28^{\circ} = -\sin (90^{\circ} - 62^{\circ}) = -\sin 62^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) = -\sin (90^{\circ} - 28^{\circ}) = -\sin 28^{\circ} = -\sin (90^{\circ} - (90^{\circ} - 62^{\circ})) =