Finding unknown angles

Using inverse sine, cosine and tangent to find an angle from two sides, rounding to the nearest degree or minute.

Inverse ratios

If $\tan\theta = \dfrac{5}{8}$, then $\theta = \tan^{-1}\left(\dfrac{5}{8}\right)$. The inverse button (often SHIFT then tan) turns a ratio back into an angle.

Example 1●●

Find $\theta$ when the opposite side is $5$ and the adjacent side is $8$.

$\tan\theta = \dfrac{5}{8}$

$\theta = \tan^{-1}(0.625) \approx 32^\circ$

Example 2●●

Find $\theta$ when the adjacent side is $7$ and the hypotenuse is $10$.

$\cos\theta = 0.7$, so $\theta \approx 45.6^\circ$

Flashcards

$ an heta = 1$

$\theta = 45^\circ$

$sin heta = 0.5$

$\theta = 30^\circ$

Minutes in a degree

$60$

Practice

Opposite $5$, adjacent $8$. Find $\theta$ to the nearest degree.
Adjacent $7$, hypotenuse $10$. Find $\theta$ to 1 d.p.
Opposite $4$, hypotenuse $9$. Find $\theta$ to 1 d.p.
Opposite and adjacent sides are both $6$. Find $\theta$.
Opposite $4$, hypotenuse $8$. Find $\theta$.
Sides $3, 4, 5$. Find the angle opposite the side of length $3$, to 1 d.p.
Adjacent $5$, hypotenuse $10$. Find $\theta$.
Write $45.57^\circ$ in degrees and minutes. How many minutes (nearest)?