It is commonly used to solve the inverse kinematics equations of robots

Find the equation: sin(θ)=asin(\theta)=asin(θ)=a

Have, cos (theta) = + 1 – a2cos (\ theta) = / PM/SQRT {1 – a ^ {2}} cos (theta) = + 1 – a2

Therefore, Theta = atan2 (sin (theta), cos (theta)) = atan2 (a, plus or minus 1 – a2) \ theta = atan2 (sin (\ theta), cos (\ theta)) = atan2 (a, / PM/SQRT {1 – a ^ {2}}) theta = atan2 (sin (theta), cos (theta)) = atan2 (a, plus or minus 1 – a2)

Find the equation: cos(θ)=bcos(\theta)=bcos(θ)=b

Have, sin (theta) = + 1 – b2sin (\ theta) = \ PM \ SQRT {1 – b ^ {2}} sin = + 1 – b2 (theta)

Therefore, Theta = atan2 (sin (theta), cos (theta)) = atan2 \ theta (+ / – 1 – b2, b) = atan2 (sin (\ theta), cos (\ theta)) = atan2 (/ PM/SQRT {1 – b ^ {2}}, B) theta = atan2 (sin (theta), cos (theta)) = atan2 (+ / – 1 – b2, b)

A equation: a ⋅ cos (theta) + b ⋅ sin (theta) = 0 a \ cdot cos (\ theta) + b \ cdot sin (\ theta) = 0 a ⋅ cos (theta) + b ⋅ sin (theta) = 0

To: A = a2 + b2sin (alpha) ⇒ sin (alpha) = aa2 + b2, b = a2 + b2cos (alpha) ⇒ cos (alpha) = ba2 + b2a = \ SQRT {^ a + b ^ {2} {2}} sin (\ alpha) \ Rightarrow sin(\alpha)=\frac{a}{\sqrt{a^{2}+b^{2}}}, B = \ SQRT {^ a + b ^ {2} {2}} cos (\ alpha) \ Rightarrow cos (\ alpha) = \ frac {b} {\ SQRT {^ a + b ^ {2} {2}}} a = a2 + b2 sin (alpha) ⇒ sin = a2 + b2 (alpha) A, b = a2 + b2 cos (alpha) ⇒ cos (alpha) = a2 + b2 b

There are, A2 + b2sin (alpha) cos (theta) + a2 + b2cos (alpha) sin (theta) = 0 \ SQRT {^ a + b ^ {2} {2}} sin (\ alpha) cos (\ theta) + \ SQRT {^ a + b ^ {2} {2}} cos (\ alpha) sin = (\ theta) 0 a2 + b2 sin (alpha) cos (theta) + a2 + b2 cos (alpha) sin (theta) = 0

That is, A2 + b2sin (alpha + theta) = 0 ⇒ sin (alpha + theta) = 0 a2 + b2 = 0 \ SQRT {^ a + b ^ {2} {2}} sin (+ \ \ alpha theta) = 0 \ Rightarrow Sin (\ alpha + \ theta) = \ frac {0} {\ SQRT {a ^ {2} + b ^ {2}}} = 0 a2 + b2 sin (alpha + theta) = 0 ⇒ sin (alpha + theta) = a2 + b2 = 0 0

So, cos (alpha + theta) = + 1-0 = + 1 cos (\ alpha + \ theta) = / PM/SQRT {1-0} = \ 1 PM cos (alpha + theta) = + 1-0 = + 1

Therefore, Theta = atan2 (sin (alpha + theta), cos (alpha + theta)) – atan2 (sin (alpha), cos (alpha)) = 0 – atan2 (a, b) = – atan2 (a, b) \ theta = atan2 (sin (\ alpha + \ theta), cos(\alpha+\theta))-atan2(sin(\alpha), cos(\alpha))=0-atan2(a, b)=-atan2(a, B) theta = atan2 (sin (alpha + theta), cos (alpha + theta)) – atan2 (sin (alpha), cos (alpha)) = 0 – atan2 (a, b) = – atan2 (a, b)

A equation: a ⋅ cos (theta) + b ⋅ sin (theta) = ca \ cdot cos (\ theta) + b \ cdot sin (\ theta) = ca ⋅ cos (theta) + b ⋅ sin (theta) = c

To: A = a2 + b2sin (alpha) ⇒ sin (alpha) = aa2 + b2, b = a2 + b2cos (alpha) ⇒ cos (alpha) = ba2 + b2a = \ SQRT {^ a + b ^ {2} {2}} sin (\ alpha) \ Rightarrow sin(\alpha)=\frac{a}{\sqrt{a^{2}+b^{2}}}, B = \ SQRT {^ a + b ^ {2} {2}} cos (\ alpha) \ Rightarrow cos (\ alpha) = \ frac {b} {\ SQRT {^ a + b ^ {2} {2}}} a = a2 + b2 sin (alpha) ⇒ sin = a2 + b2 (alpha) A, b = a2 + b2 cos (alpha) ⇒ cos (alpha) = a2 + b2 b

There are, A2 + b2sin (alpha) cos (theta) + a2 + b2cos (alpha) sin (theta) = c \ SQRT {^ a + b ^ {2} {2}} sin (\ alpha) cos (\ theta) + \ SQRT {^ a + b ^ {2} {2}} cos (\ alpha) sin = (\ theta) Ca2 + b2 sin (alpha) cos (theta) + a2 + b2 cos (alpha) sin (theta) = c

That is, A2 + b2sin (alpha + theta) = c ⇒ sin (alpha + theta) = ca2 + b2 \ SQRT {^ a + b ^ {2} {2}} sin (\ alpha + \ theta) = c \ Rightarrow Sin (\ alpha + \ theta) = \ frac {c} {\ SQRT {a ^ {2} + b ^ {2}}} a2 + b2 sin (alpha + theta) = c ⇒ sin (alpha + theta) = a2 + b2 c

So, Cos (alpha + theta) = + a2 + b2 – c2a2 + b2cos (+ \ \ alpha theta) = \ PM \ frac {\ SQRT {^ a + b ^ {2} {2} – c ^ {2}}} {\ SQRT {a ^ {2} + b ^ {2}}} cos (alpha + theta) = + a2 + b2 A2 + b2 – c2

Therefore, Theta = atan2 (sin (alpha + theta), cos (alpha + theta)) – atan2 (sin (alpha), cos (alpha)) = atan2 (c + a2 + b2 – c2) – atan2 (a, b) \ theta = atan2 (sin (\ alpha + \ theta), cos(\alpha+\theta)) – atan2(sin(\alpha), cos(\alpha))= atan2(c, \pm \sqrt{a^{2}+b^{2}-c^{2}}) – atan2(a, B) theta = atan2 (sin (alpha + theta), cos (alpha + theta)) – atan2 (sin (alpha), cos (alpha)) = atan2 (c + a2 + b2 – c2) – atan2 (a, b)