Question
Question
and
. Find the value of
by using a sketch and
not finding the value of
.
not finding the value of
Ask by Norris Warren. in South Africa
Jan 24,2025
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To solve
, we can rearrange it to find the sine and cosine values. First,
. Since
, we are looking at the second quadrant, where sine is positive and cosine is negative.
Using the identity
, we can first find
and
:
- Set
and , where is a positive scaling factor. - Then
, which is consistent with our quadrant determination. - Now using
, we have . - Solving this gives
or . - Thus,
.
Now, substituting back, we find:
and
.
Finally, for
:
So, the value of
is
.
Now, as a side note, it’s interesting to think about where this all fits historically. The study of trigonometric functions dates back to ancient civilizations, with mathematicians like Hipparchus and later Islamic scholars developing tables and understanding of these ratios. They utilized geometry for astronomy and built a foundation for calculus and analysis!
On the practical side, understanding these trigonometric ratios and their signs in different quadrants can significantly simplify physics problems, especially in mechanics where angles of projection or forces need resolution into components. So mastering this concept is not just for math class; it’s essential across many real-world scenarios!