Esimerkki

Ratkaise                           asteina

\sin \alpha = 0,30
sinα=0,30\sin \alpha = 0,30
\sin \alpha = 0,30 \quad ||\sin^{-1}
sinα=0,30∣∣sin−1\sin \alpha = 0,30 \quad ||\sin^{-1}

Ratkaisu

\alpha= 17,45\ldots^{\circ}
α=17,45…∘\alpha= 17,45\ldots^{\circ}
\alpha = 180^{\circ}-17,45\ldots^{\circ}+ n \cdot 360^{\circ}
α=180∘−17,45…∘+n⋅360∘\alpha = 180^{\circ}-17,45\ldots^{\circ}+ n \cdot 360^{\circ}

Määritetään kaikki ratkaisut

\alpha = 17,45\ldots^{\circ}+ n \cdot 360^{\circ}
α=17,45…∘+n⋅360∘\alpha = 17,45\ldots^{\circ}+ n \cdot 360^{\circ}

tai

\alpha = 162,54 \ldots^{\circ}+ n \cdot 360^{\circ}
α=162,54…∘+n⋅360∘\alpha = 162,54 \ldots^{\circ}+ n \cdot 360^{\circ}
\alpha = 17,45\ldots^{\circ}+ n \cdot 360^{\circ}
α=17,45…∘+n⋅360∘\alpha = 17,45\ldots^{\circ}+ n \cdot 360^{\circ}

tai

\alpha \approx 17,5^{\circ}+ n \cdot 360^{\circ}
α≈17,5∘+n⋅360∘\alpha \approx 17,5^{\circ}+ n \cdot 360^{\circ}

tai

\alpha\approx 162,5 ^{\circ}+ n \cdot 360^{\circ}
α≈162,5∘+n⋅360∘\alpha\approx 162,5 ^{\circ}+ n \cdot 360^{\circ}

Vastaus:

jossa n on kokonaisluku.

Esimerkki

Ratkaise                        asteina

2\cos 3\alpha = 1
2cos3α=12\cos 3\alpha = 1

Ratkaisu

2\cos 3\alpha = 1 \quad \ \ \ \ ||:2
2cos3α=1    ∣∣:22\cos 3\alpha = 1 \quad \ \ \ \ ||:2
\cos 3\alpha = 0,5 \quad ||\cos^{-1}
cos3α=0,5∣∣cos−1\cos 3\alpha = 0,5 \quad ||\cos^{-1}
3\alpha = 60^{\circ}
3α=60∘3\alpha = 60^{\circ}

Määritetään kaikki ratkaisut

3\alpha = 60^{\circ}+ n \cdot 360^{\circ}
3α=60∘+n⋅360∘3\alpha = 60^{\circ}+ n \cdot 360^{\circ}
3\alpha = -60^{\circ}+ n \cdot 360^{\circ}
3α=−60∘+n⋅360∘3\alpha = -60^{\circ}+ n \cdot 360^{\circ}
\alpha = 20^{\circ}+ n \cdot 120^{\circ}
α=20∘+n⋅120∘\alpha = 20^{\circ}+ n \cdot 120^{\circ}
\alpha = -20^{\circ}+ n \cdot 120^{\circ}
α=−20∘+n⋅120∘\alpha = -20^{\circ}+ n \cdot 120^{\circ}
\alpha = \pm 20^{\circ}+ n \cdot 120^{\circ}
α=±20∘+n⋅120∘\alpha = \pm 20^{\circ}+ n \cdot 120^{\circ}

tai

tai

, jossa n on kokonaisluku.

Vastaus:

Esimerkki

Ratkaise                    radiaaneina

\sin x = 1
sinx=1\sin x = 1

Ratkaisu

\sin x = 1 \quad \ \ \ \ ||\sin^{-1}
sinx=1    ∣∣sin−1\sin x = 1 \quad \ \ \ \ ||\sin^{-1}
x = \dfrac{\pi}{2}
x=π2x = \dfrac{\pi}{2}

Määritetään kaikki ratkaisut

x = \dfrac{\pi}{2}+n \cdot 2 \pi
x=π2+n⋅2πx = \dfrac{\pi}{2}+n \cdot 2 \pi
x =\pi - \dfrac{\pi}{2}+n \cdot 2 \pi
x=π−π2+n⋅2πx =\pi - \dfrac{\pi}{2}+n \cdot 2 \pi

tai

x = \dfrac{\pi}{2}+n \cdot 2 \pi
x=π2+n⋅2πx = \dfrac{\pi}{2}+n \cdot 2 \pi
x = \dfrac{\pi}{2}+n \cdot 2 \pi
x=π2+n⋅2πx = \dfrac{\pi}{2}+n \cdot 2 \pi

tai

Vastaus:

x = \dfrac{\pi}{2} + n \cdot 2 \pi
x=π2+n⋅2πx = \dfrac{\pi}{2} + n \cdot 2 \pi

, jossa n on kokonaisluku.

Esimerkki

Ratkaise                        radiaaneina

\sin \alpha = -\frac{1}{2}
sinα=−12\sin \alpha = -\frac{1}{2}

Ratkaisu

\sin \alpha = -\dfrac{1}{2} \quad \ \ \ \ ||\sin^{-1}
sinα=−12    ∣∣sin−1\sin \alpha = -\dfrac{1}{2} \quad \ \ \ \ ||\sin^{-1}
\alpha = -\dfrac{\pi}{6}
α=−π6\alpha = -\dfrac{\pi}{6}

Määritetään kaikki ratkaisut

\alpha = -\dfrac{\pi}{6}+n \cdot 2 \pi
α=−π6+n⋅2π\alpha = -\dfrac{\pi}{6}+n \cdot 2 \pi
\alpha =\pi - \Big(-\dfrac{\pi}{6}\Big)+n \cdot 2 \pi
α=π−(−π6)+n⋅2π\alpha =\pi - \Big(-\dfrac{\pi}{6}\Big)+n \cdot 2 \pi

tai

\alpha = -\dfrac{\pi}{6}+n \cdot 2 \pi
α=−π6+n⋅2π\alpha = -\dfrac{\pi}{6}+n \cdot 2 \pi
\alpha =\dfrac{6\pi}{6}+\dfrac{\pi}{6} +n \cdot 2 \pi
α=6π6+π6+n⋅2π\alpha =\dfrac{6\pi}{6}+\dfrac{\pi}{6} +n \cdot 2 \pi

tai

Vastaus:

\alpha = -\dfrac{\pi}{6} + n \cdot 2 \pi \text{ tai } \alpha=\dfrac{7 \pi}{6}+n \cdot 2 \pi
α=−π6+n⋅2π tai α=7π6+n⋅2π\alpha = -\dfrac{\pi}{6} + n \cdot 2 \pi \text{ tai } \alpha=\dfrac{7 \pi}{6}+n \cdot 2 \pi

, jossa n on kokonaisluku.

Esimerkki

Ratkaise                                      asteina

\tan (2\alpha-30^{\circ}) = 0,30
tan(2α−30∘)=0,30\tan (2\alpha-30^{\circ}) = 0,30

Ratkaisu

\tan (2\alpha-30^{\circ}) = 0,30 \quad \ \ \ \ ||\tan^{-1}
tan(2α−30∘)=0,30    ∣∣tan−1\tan (2\alpha-30^{\circ}) = 0,30 \quad \ \ \ \ ||\tan^{-1}
2 \alpha -30^{\circ} = 16,69 \ldots^{\circ}
2α−30∘=16,69…∘2 \alpha -30^{\circ} = 16,69 \ldots^{\circ}

Määritetään kaikki ratkaisut

\alpha \approx 23,3^{\circ} + n \cdot 180^{\circ}
α≈23,3∘+n⋅180∘\alpha \approx 23,3^{\circ} + n \cdot 180^{\circ}

Vastaus:

\alpha \approx 23,3^{\circ}+n \cdot 180^{\circ}
α≈23,3∘+n⋅180∘\alpha \approx 23,3^{\circ}+n \cdot 180^{\circ}

, jossa n on kokonaisluku.

2 \alpha = 46,69 \ldots^{\circ}
2α=46,69…∘2 \alpha = 46,69 \ldots^{\circ}
\alpha = 23,34 \ldots^{\circ}
α=23,34…∘ \alpha = 23,34 \ldots^{\circ}