What's the difference between sin inverse and 1 over sin?

Sin inverse finds an angle from a ratio — so sin inverse of 0.5 equals 30 degrees. 1 over sin is just 1 divided by the sine of an angle (it's called the cosecant). They are completely different functions.

Can the answer be more than 90 degrees?

Not in a right triangle. The two acute angles add to 90 degrees, so each must be less than 90 degrees. If your inverse-trig answer comes out greater than 90 degrees or negative, something's gone wrong — usually the ratio was flipped (opposite/adjacent swapped) or the calculator is in the wrong mode.

What's the difference between sin inverse and 1 over sin?

Sin inverse finds an angle from a ratio — so sin inverse of 0.5 equals 30 degrees. 1 over sin is just 1 divided by the sine of an angle (it's called the cosecant). They are completely different functions.

Can the answer be more than 90 degrees?

Not in a right triangle. The two acute angles add to 90 degrees, so each must be less than 90 degrees. If your inverse-trig answer comes out greater than 90 degrees or negative, something's gone wrong — usually the ratio was flipped (opposite/adjacent swapped) or the calculator is in the wrong mode.

Year 11 General Applications Of Trigonometry

Finding An Angle In A Right-Angled Triangle

Q: What is sin inverse on a calculator?

It's the inverse sine function, sometimes written as arcsin. On most calculators you press SHIFT (or 2nd) and then the sin key. It takes a ratio between -1 and 1 and returns the angle whose sine equals that ratio.

Q: How do I convert decimal degrees to minutes?

Multiply the decimal part by 60. For example, 23.578 degrees has decimal part 0.578, and 0.578 times 60 equals 34.68, so 23.578 degrees is approximately 23 degrees 35 minutes. Most calculators have a D-M-S button that does this in one keystroke.

Q: How many minutes are in a degree?

Sixty: 1 degree equals 60 minutes. And each minute splits into 60 seconds (1 minute equals 60 seconds), so 1 degree equals 3600 seconds. Most school problems only go to the nearest minute.

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Theory

If two sides of a right-angled triangle are known, the unknown angle can be found using the inverse trig functions $\sin^{- 1}$ , $\cos^{- 1}$ or $\tan^{- 1}$ . Pick the trig ratio that uses the two known sides, then apply its inverse to get the angle in degrees.

To find an unknown angle in a right-angled triangle you need:

any two known sides (no angle needed)
the inverse trig functions on your calculator: $\sin^{- 1}$ , $\cos^{- 1}$ , $\tan^{- 1}$ (usually accessed via SHIFT or 2nd above sin, cos and tan)

The inverse trig functions reverse the trig ratios — they take a ratio and give back the angle:

If $\sin θ = k$ , then $θ = \sin^{- 1} (k)$ .
If $\cos θ = k$ , then $θ = \cos^{- 1} (k)$ .
If $\tan θ = k$ , then $θ = \tan^{- 1} (k)$ .

The calculator gives an answer in degrees — round either to the nearest degree, or for finer accuracy to the nearest minute ( $1^{\circ} = 60^{'}$ ).

Two known sides → use

\tan^{- 1}

θ = \tan^{- 1} (\frac{3}{4})

Opp & hyp known → use

\sin^{- 1}

θ = \sin^{- 1} (\frac{4.5}{5})

The three inverse trig functions

θ = \sin^{- 1} (\frac{Opp}{Hyp}) θ = \cos^{- 1} (\frac{Adj}{Hyp}) θ = \tan^{- 1} (\frac{Opp}{Adj})

θ = \sin^{- 1} (\frac{Opp}{Hyp}), θ = \cos^{- 1} (\frac{Adj}{Hyp}), θ = \tan^{- 1} (\frac{Opp}{Adj})

Converting decimal degrees to minutes

One degree equals $60$ minutes: $1^{\circ} = 60^{'}$ . To convert a decimal answer like ${23.578}^{\circ}$ :

0.578 \times 60 = 34.68 \Rightarrow {23.578}^{\circ} \approx 23^{\circ} 35^{'}

23.578 ° \approx 23 ° 35'

Most calculators have a D-M-S or ° ' '' button that does this automatically.

          The inverse buttons are not reciprocals: sin−1 is NOT the same as 1sin. sin−1 finds an angle from a ratio; 1sin is one divided by sine. Use the SHIFT/2nd key, not the reciprocal key.
        

How to find any unknown angle

Label the three sides relative to the unknown angle $θ$ : Opposite, Adjacent, Hypotenuse.
Identify the two sides whose lengths you know.
Choose the trig ratio that uses those two sides — sin (Opp + Hyp), cos (Adj + Hyp), or tan (Opp + Adj) — and write $\sin θ$ , $\cos θ$ or $\tan θ$ equal to the fraction.
Apply the inverse trig function to both sides to get $θ$ , then round as instructed.

Example 1 — Find angle from sin

Find

θ

\sin θ = 0.84

(nearest degree).

Solution

Apply $\sin^{- 1}$ to both sides:

$θ$	$=$	$\sin^{- 1} (0.84)$
$θ$	$=$	$57.14 \dots^{\circ}$
$θ$	$\approx$	$57^{\circ}$

θ = \sin^{- 1} (0.84) \approx 57 °

Example 2 — Two given sides

A right triangle has the side opposite

θ

of length

3

and the side adjacent of length

4

. Find

θ

(nearest degree).

Solution

Sketch the triangle:

Opposite and adjacent are known, so use $\tan$ :

$\tan θ$	$=$	$\frac{3}{4}$
$θ$	$=$	$\tan^{- 1} (\frac{3}{4})$
$θ$	$\approx$	$37^{\circ}$

θ = \tan^{- 1} (\frac{3}{4}) \approx 37 °

Example 3 — Ladder angle

5

m ladder leans against a wall, reaching

4.5

m up. Find the angle the ladder makes with the ground (nearest degree).

Solution

Sketch: ladder is the hypotenuse, height is opposite:

$\sin θ$	$=$	$\frac{4.5}{5} = 0.9$
$θ$	$=$	$\sin^{- 1} (0.9)$
$θ$	$\approx$	$64^{\circ}$

θ = \sin^{- 1} (0.9) \approx 64 °

Example 4 — Nearest minute

Find

θ

\tan θ = 3

(nearest minute).

Solution

Apply $\tan^{- 1}$ , then convert the decimal part to minutes:

$θ$	$=$	$\tan^{- 1} (3)$
$θ$	$=$	$71.565 \dots^{\circ}$
$θ$	$=$	$71^{\circ} {33.9}^{'}$
$θ$	$\approx$	$71^{\circ} 34^{'}$

θ = \tan^{- 1} (3) \approx 71 ° 34'

Common pitfalls

$\sin^{- 1}$ is not $\frac{1}{\sin}$ . The inverse buttons (SHIFT + sin/cos/tan) find an angle from a ratio. The reciprocal buttons (

1 / \sin

) just do one divided by the ratio. They give wildly different answers.

Answer out of range. An angle in a right triangle is always between

0^{\circ}

and

90^{\circ}

. If your calculator gives a negative number or above

90^{\circ}

, check whether you flipped the fraction or hit the wrong button.

Wrong calculator mode. Set the calculator to degree mode (DEG) before any trig calculation. Radians would give an answer like

0.96

instead of

57^{\circ}

for

\sin^{- 1} (0.84)

Use brackets for fractions. For

\sin^{- 1} (\frac{1}{\sqrt{3}})

, type SHIFT sin ( 1 / √ 3 ). Missing brackets can change the order of operations and your answer.

Frequently asked questions

How do I find an angle in a right triangle?

Pick the trig ratio that uses the two sides you know, write $\sin θ$ , $\cos θ$ or $\tan θ$ equal to the fraction, then apply the inverse function ( $\sin^{- 1}$ , $\cos^{- 1}$ or $\tan^{- 1}$ ) on your calculator.

What is

\sin^{- 1}

on a calculator?

It's the inverse sine function, sometimes written as $\arcsin$ . On most calculators you press SHIFT (or 2nd) and then the sin key. It takes a ratio between $- 1$ and $1$ and returns the angle whose sine equals that ratio.

What's the difference between

\sin^{- 1}

and

\frac{1}{\sin}

$\sin^{- 1}$ finds an angle from a ratio — so $\sin^{- 1} (0.5) = 30^{\circ}$ . $\frac{1}{\sin}$ is just $1$ divided by the sine of an angle (it's called the cosecant). They are completely different functions.

How do I convert decimal degrees to minutes?

Multiply the decimal part by $60$ . For example, ${23.578}^{\circ}$ has decimal part $0.578$ , and $0.578 \times 60 = 34.68$ , so ${23.578}^{\circ} \approx 23^{\circ} 35^{'}$ . Most calculators have a D-M-S or ° ' '' button that does this in one keystroke.

How many minutes are in a degree?

Sixty: $1^{\circ} = 60^{'}$ . And each minute splits into $60$ seconds ( $1^{'} = 60^{″}$ ), so $1^{\circ} = 3600^{″}$ . Most school problems only go to the nearest minute.

Can the answer be more than

90^{\circ}

Not in a right triangle. The two acute angles add to $90^{\circ}$ , so each must be less than $90^{\circ}$ . If your inverse-trig answer comes out greater than $90^{\circ}$ or negative, something's gone wrong — usually the ratio was flipped (opposite/adjacent swapped) or the calculator is in the wrong mode.

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Applications Of Right-Angled Triangles To Problem-Solving

Finding An Angle In A Right-Angled Triangle

📖 Theory

The three inverse trig functions

Converting decimal degrees to minutes

How to find any unknown angle

Common pitfalls

Frequently asked questions

Theory