<    1    >

---

Inverse cosine

The inverse cosine can be expressed with logarithms as

 

---

Explanation

We start with the cosine

Substitution of e^iθ = k gives

We only continue with the positive solution and substitute k = e^iθ so that

On both sides we take the logarithm

Substitution of θ = arccos x gives

 

---

Example 1

You can see that cos⁻¹(1) = 0, as

cos⁻¹(1) = −i ln (1 + √0) = −i ln 1 = 0

 

---

Example 2

You can see that cos⁻¹(−1) = π = 3,141592..., as

cos⁻¹(−1) = −i ln (−1 + √0) = −i ln (−1) =  −i ·π i =  −i² · π = π

because ln (−1) = πi.

 

---

Example 3

You can see that cos⁻¹(0) = 1,57079632…, as

cos⁻¹(0) = −i ln (√−1) = −i ln (i) =  −i · ½π i = ½π

because ln (i) = ½π i.

 

---

Deutsch   Español   Français   Nederlands   中文