How much trigonometry is there in Calcuel 2
Trigonometric quantities in right triangles
Cosine and sine play a key role in studying right triangles. Given a right triangle ABC with side lengths a, b, c and angles α, β, γ with γ = π / 2, we can assume by shifting, rotating and mirroring that A is the zero point, C on the positive x-axis and B is in the first quadrant. Then B = (b, a) is a point on the circle K.cwhose radius corresponds to the hypotenuse c of the triangle ABC. Hence applies
(b, a) = B = c (cos α, sin α), b = c cos α, a = c sin α.
The other trigonometric functions tangent, cotangent, secant and Kosenkan can also be illustrated with the help of right triangles. But first we want the addition theorems
cos (α + β) = cos α cos β - sin α sin β,
sin (α + β) = cos α sin β + sin α cos β for all α, β ∈ ℝ
prove with the help of right triangles.
Second proof of addition theorems
Let α, β ∈ ℝ. We first assume that α, β> 0 and α + β <π 2.="" everything="" else="" will="" follow="" from="" this.="" under="" these="" conditions,="" α="" +="" β="" is="" the="" angle="" of="" a="" right="" triangle.="" we="" construct="" such="" a="" triangle="" as="" follows,="" where="" with="" ∠="" abc="" we="" denote="" the="" angle="" of="" a="" triangle="" with="" corners="" a,="" b,="" c="" belonging="" to="" corner="">π>
Let P = (cos α, sin α). Let Q = (x0, 0) the point on the x-axis with ∠0QP = β. Then ∠Q0P = α. Finally, let A be the point on the straight line QP with ∠0AP = π / 2. Then ∠0PA = α + β. So the triangle 0PA is as desired.
We can now count on:
|sin (α + β)||= 0A = x0 sin β|
|= (cos α + cot β sin β) sin β|
|= sin α cos β + cos α sin β.|
The cosine theorem is proven analogously (or deduced from the sine theorem with the help of Pythagoras' theorem).
We now consider more general angles. The addition theorems are clear because of sin 0 = cos (π / 2) = 0 and sin (π / 2) = cos 0 = 1, if one of the two angles is 0 or π / 2. If α, β ∈ [ 0, π / 2] with α + β ≥ π / 2, then let α * = π / 2 - α, β * = π / 2 - β. Then according to what has already been proven that
|sin (α + β)||= sin (π - (α + β))|
|= sin (α * + β *)|
|= sin α * cos β * + cos α * sin β *|
|= cos α sin β + sin α cos β|
|= sin α cos β + cos α sin β.|
We argue analogously for the cosine. We have thus shown the addition theorems for all α, β ∈ [0, π / 2] (this corresponds to points P and Q on the unit circle in the first quadrant). From this we can deduce the theorems for any angles α, β ∈ ℝ by adding
α = α1 + k1 π / 2, β = β1 + k2 π / 2 with k1, k2 ∈ ℤ and α1, β1 ∈ [0, π / 2]
write and apply the shift formulas.
Identification of the trigonometric quantities
The six trigonometric quantities
cos α, sin α, tan α, cot α, sec α, csc α
appear in numerous geometric figures. We consider two typical examples. The identification of the quantities results from their definition by applying the ray theorem.
In the following we denote the length of the line between two points A and B of the plane with AB. If A = (x1, y1) and B = (x2, y2), so it is true
In our first figure we consider a point P on the unit circle with the angle α ∈] 0, π / 2 [and the half-ray of the plane defined by the zero point 0 and P. This half-ray intersects the Cartesian coordinate grid at two points R and S. All six quantities are included in the resulting figure.
With the help of the figure we can illustrate many properties of the functions. For example, tan α = TR tends towards infinity if the angle α tends towards π / 2. Furthermore, R = S = (1, 1) and thus tan α = cot α, if α = π / 4.
In the second figure we consider the tangent of the unit circle K at P. It defines two points on the axes:
Combined, we get the following figure, which focuses on the four perhaps less obvious quantities tan α, cot α, sec α and csc α:
The figure can be analyzed further. The intersection of the two tangent lines drawn in green defines, for example, the bisector of α.
Our considerations also motivate the naming of the trigonometric functions tan, cot, sec and csc. The tangent and the cotangent measure the lengths of certain sections of tangents of the unit circle. Likewise, the secant and the cosecant measure the lengths of certain sections of secants of the unit circle. A “co” always refers to the complementary angle π / 2 - α. This also explains why the function defined by 1 / cos α is not referred to as “co” even though the cosine is used. In our triangles, the quantity 1 / cos α is assigned to the angle α, while 1 / sin α belongs to the complementary angle π / 2 - α.
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