SOLVE SINA
You might lượt thích to read about Trigonometry first!
Right Triangle
The Trigonometric Identities are equations that are true for Right Angled Triangles. (If it is not a Right Angled Triangle go lớn the Triangle Identities page.)
Each side of a right triangle has a name:
Adjacent is always next to the angle
And Opposite is opposite the angle
We are soon going to lớn be playing with all sorts of functions, but remember it all comes back to that simple triangle with:
Angle θHypotenuseAdjacentOppositeSine, Cosine và Tangent
The three main functions in trigonometry are Sine, Cosine và Tangent.
They are just the length of one side divided by another
For a right triangle with an angle θ :
| tan(θ) = Opposite / Adjacent |
For a given angle θ each ratio stays the same no matter how big or small the triangle is
When we divide Sine by Cosine we get:
sin(θ)cos(θ) = Opposite/HypotenuseAdjacent/Hypotenuse = OppositeAdjacent = tan(θ)
So we can say:
That is our first Trigonometric Identity.
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Cosecant, Secant and Cotangent
We can also divide "the other way around" (such as Adjacent/Opposite instead of Opposite/Adjacent):
Pythagoras TheoremFor the next trigonometric identities we start with Pythagoras" Theorem:
Dividing through by c2 gives a2c2 + b2c2 = c2c2 This can be simplified to: (ac)2 + (bc)2 = 1 Now, a/c is Opposite / Hypotenuse, which is sin(θ) And b/c is Adjacent / Hypotenuse, which is cos(θ) So (a/c)2 + (b/c)2 = 1 can also be written: Note:sin2 θ means khổng lồ find the sine of θ, then square the result, andsin θ2 means to lớn square θ, then vì chưng the sine function Example: 32°Using 4 decimal places only: sin(32°) = 0.5299...cos(32°) = 0.8480...Xem thêm: Hãy Tưởng Tượng Em Gặp Người Lính Trong Bài Đồng Chí, Just A Moment Now let"s calculate sin2 θ + cos2 θ: 0.52992 + 0.84802 = 0.2808... + 0.7191... = 0.9999... We get very close to lớn 1 using only 4 decimal places. Try it on your calculator, you might get better results! sin2 θ = 1 − cos2 θcos2 θ = 1 − sin2 θtan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1cot2 θ + 1 = csc2 θcot2 θ = csc2 θ − 1
But Wait ... There is More!There are many more identities ... Here are some of the more useful ones: Opposite Angle Identitiessin(−θ) = −sin(θ) cos(−θ) = cos(θ) tan(−θ) = −tan(θ) Double Angle Identities
Half Angle IdentitiesNote that "±" means it may be either one, depending on the value of θ/2
Angle Sum và Difference IdentitiesNote that means you can use plus or minus, and the means lớn use the opposite sign. sin(A B) = sin(A)cos(B) cos(A)sin(B) cos(A B) = cos(A)cos(B) sin(A)sin(B) tan(A B) = tan(A) tan(B)1 tan(A)tan(B) cot(A B) = cot(A)cot(B) 1cot(B) cot(A) Triangle IdentitiesThere are also Triangle Identities which apply to all triangles (not just Right Angled Triangles) |
