# HOW DO YOU PROVE SIN^2X + COS^2X = 1? + EXAMPLE

to lớn start off, I understand the proof behind this identity, & I can visualize it in my head with the unit circle.

Bạn đang xem: How do you prove sin^2x + cos^2x = 1? + example

They only need to remember three facts – that \$sin 30^circ = ½\$ , that \$ an 45^circ =1\$, và that \$sin^2x + cos^2x =1\$ . Just about everything else they need khổng lồ know about trigonometry can be derived from these.

and I realized I don"t have a complete understanding on practical use of the identity. Therefore I am looking for an explanation and some practical examples on why it is so important.

giới thiệu
Cite
Follow
edited Jul 8, 2013 at 7:57 Tobias Kienzler
asked Jul 8, 2013 at 7:39 user22979user22979
\$endgroup\$
2
showroom a bình luận |

Sorted by: Reset to mặc định
Highest score (default) Date modified (newest first) Date created (oldest first)
11
\$egingroup\$
The "special angles" in trig are \$0^circ\$, \$30^circ\$, \$45^circ\$, \$60^circ\$, and \$90^circ\$ (and their counterparts in quadrants II, III, và IV).

Presumably, the sines and cosines at \$0^circ\$ và \$90^circ\$ are obvious to lớn the quoted author, so the quotation boils-down what"s needed to lớn remember values at the remaining angles to lớn three facts.

Xem thêm: Khi Nào Dùng Do Does Khi Nào Dùng Do Vậy? Câu Hỏi 62094 Https://Hoidap247

"\$sin^2 heta+cos^2 heta=1\$" allows for conversion between sines and cosines. Always handy.

"\$sin 30^circ = 1/2\$" immediately gives \$cos 60^circ =1/2\$. (Complementary angles are cool lượt thích that, since "cosine" means "complementary sine".) With the help of the Pythagorean identity above, we also get \$cos 30^circ = sin 60^circ = sqrt3/2\$. (If you"re going khổng lồ commit any of these khổng lồ memory, it makes sense to lớn choose the simple fraction, \$1/2\$. Personally, I think remembering "\$cos 60^circ = 1/2\$" is easier, because I can "see" it more easily in dropping a perpendicular from the vị trí cao nhất vertex of an equilateral triangle.)

"\$ an 45^circ=1\$" encodes information about the remaining special angle, saying that (in Quadrant I) \$sin 45^circ = cos 45^circ\$. Again with the help of the Pythagorean identity, we have \$1 = sin^2 45^circ + cos^2 45^circ = 2 sin^2 45^circ\$, so that \$sin 45^circ = cos 45^circ = 1/sqrt2\$.

So, the three facts help lớn recover a total of seven facts. & then getting the remaining secants, tangents, cosecants, & cotangents is a simple matter of using various ratios. The sine-cosine Pythagorean identity also gives rise khổng lồ the others, by dividing-through by \$cos^2 heta\$ or \$sin^2 heta\$ ...

\$\$eginalignfracsin^2 hetacos^2 heta + fraccos^2 hetacos^2 heta = frac1cos^2 heta quad & o quad an^2 heta + 1 = sec^2 heta \<6pt>fracsin^2 hetasin^2 heta + fraccos^2 hetasin^2 heta = frac1sin^2 heta quad & o quad 1 + cot^2 heta = csc^2 hetaendalign\$\$

... Reducing pressure lớn memorize all three relations.

Xem thêm: 2022 Nên Mua Máy Hàn Điện Tử Hãng Nào Tốt, Top 5 Máy Hàn Điện Tử Tốt Nhất Hiện Nay 2022

There are other (and better?) ways to lớn remember this stuff, but the basic point is that there"s a good deal of informational redundancy in trig. While it may seem lượt thích there are a zillion different things khổng lồ memorize, it doesn"t take long lớn realize that one can focus on a (very) few key facts và re-derive the rest on demand.