Sin 47 Degrees — Value of sin(47°) and How to Find It

Sin 47 Degrees — Value of sin(47°) and How to Find It

TL;DR

The value of sin 47 degrees is approximately 0.7314 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find ( \sin 47° ) honestly (calculator, the cofunction ( \cos 43° ), and table interpolation), gives the radian form, and places it on the unit circle.

The value of sin 47 degrees is approximately 0.7314 (0.73135370 to eight places). Unlike ( \sin 30° ) or ( \sin 45° ), the angle 47° is not a special angle, so ( \sin 47° ) has no simple exact surd — it is read from a calculator, a trig table, or rewritten as the cofunction ( \cos 43° ).

Quick Answer: Result: ( \sin 47° \approx 0.7314 ) In radians: ( \sin\left(\frac{47\pi}{180}\right) = \sin(0.82031) \approx 0.7314 ) Notation: decimal approximation — 0.73135370 (8 dp) Method shown: calculator (degree mode), the cofunction identity ( \sin 47° = \cos 43° ), and table interpolation Exact form: none simple — 47° is not a special angle, so no clean radical exists

Quick Reference — Sine Near 47 Degrees

Sin 47° sits between the special landmarks ( \sin 45° ) and ( \sin 60° ). The table below places it among its neighbours.

Angle (degrees) Angle (radians) ( \sin \theta ) Special angle?
30° ( \frac{\pi}{6} ) 0.5000 Yes (exact 12)
45° ( \frac{\pi}{4} ) 0.7071 Yes (( \frac{\sqrt{2}}{2} ))
46° ( \frac{23\pi}{90} ) 0.7193 No
47° ( \frac{47\pi}{180} ) 0.7314 No — decimal only
48° ( \frac{4\pi}{15} ) 0.7431 No
60° ( \frac{\pi}{3} ) 0.8660 Yes (( \frac{\sqrt{3}}{2} ))

The nearest exact landmark is ( \sin 45° = \frac{\sqrt{2}}{2} \approx 0.7071 ), and ( \sin 47° ) sits just 0.0243 above it.

What Does Sin 47 Degrees Mean?

Sine of an angle on the unit circle (radius 1, centered at the origin) is the y-coordinate of the point at that angle, where every point is (( \cos \theta, \sin \theta ). Here a quadrant is one of the four regions the axes cut the plane into, numbered anticlockwise from the top right; 47° lands in Quadrant I, where both coordinates are positive.

At 47° the radius has turned just past the 45° diagonal, so the point's height is a little above 0.7071. That height — about 0.7314 — is ( \sin 47° ).

How Do You Find the Value of Sin 47 Degrees?

Because 47° is not a special angle, there is no surd to simplify to. So how do you find sin 47 degrees without a calculator? You rewrite it as a cofunction or read it off a table — here are the three honest routes.

Method 1: Calculator (set to degree mode)

Type ( \sin(47) ) with the calculator in DEG mode. ( \sin 47° = 0.73135370… \approx 0.7314 ) In radian mode the same keystrokes give ( \sin(47 , \text{rad}) \approx 0.1236 ) — a completely different number, so the mode matters.

Method 2: Cofunction identity

Sine and cosine are cofunctions: ( \sin \theta = \cos(90° - \theta) ). ( \sin 47° = \cos(90°-47°) = \cos 43° ) So ( \sin 47° ) and ( \cos 43° ) are the same number, 0.7314. This is useful when a table or problem gives you cosines but you need a sine.

Method 3: Table interpolation

If a trig table lists ( \sin 45° = 0.7071 ) and ( \sin 50° = 0.7660 ), estimate ( \sin 47° ) by linear interpolation: ( \sin 47° \approx 0.7071 + \frac{47 - 45}{50 - 45}(0.7660 - 0.7071) = 0.7071 + 0.4(0.0589) = 0.7307 ) That lands within 0.001 of the true 0.7314 — close, though interpolation always carries a small error because the sine curve bends slightly between the table rows.

What is sin 47 degrees in radians?

The angle converts to ( \frac{47\pi}{180} \approx 0.8203 ) rad, but the value of the sine is the same number, ( 1 ext{approximately} 0.7314 ). Converting the angle to radians does not change the sine; it only changes how the angle is labelled.

Examples Using Sin 47 Degrees

Example 1

State ( \sin 47° ) to four decimal places. From a calculator in degree mode, ( \sin 47° = 0.7314 ).

Example 2 (wrong path first)

Find ( \sin 47° ) using a cofunction. Wrong attempt. A student writes ( \sin 47° = \sin(90° - 47°) = \sin 43° ). Why it breaks. The cofunction of sine is cosine, not sine: ( \sin \theta = \cos(90° - \theta) ). Writing ( \sin 43° ) gives 0.6820, not 0.7314 — the wrong value. Correct. ( \sin 47° = \cos(90°-47°) = \cos 43° = 0.7314 ).

Example 3

A wire runs from the top of a 10 m pole to the ground, making a 47° angle with the wire's straight length. How high is the attachment if the wire is 10 m long? Height = 10 × ( \sin 47° = 10 × 0.7314 = 7.314 ) m.

Example 4

Compare ( \sin 47° ) with ( \sin 45° ). ( \sin 45° = 0.7071 ); ( \sin 47° = 0.7314 ). The extra 2° raises the value by 0.0243, because sine is still climbing steeply near 45°.

Example 5

Verify ( \sin 47°=\cos 43° ) on a calculator. ( \sin 47° = 0.73135 ) and ( \cos 43° = 0.73135 ) — identical, confirming the cofunction identity.

Sin 47 Degrees — Where Things Go Sideways

Most errors on a non-special angle come from a few repeatable habits, not from the arithmetic.

Mistake 1: Using the wrong cofunction

Where it slips in: rewriting ( \sin 47° ) as a complementary angle and keeping the same function. Don't do this: writing ( \sin 47° = \sin 43° ). The correct way: the complement of sine is cosine — ( \sin 47° = \cos 43° ).

Mistake 2: Hunting for an exact surd

Where it slips in: assuming every angle has a clean value like ( \sin 45°=\frac{\sqrt{2}}{2} ). Don't do this: trying to write ( \sin 47° ) as a simple radical.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: the calculator was left in radian mode. Don't do this: reporting ( \sin(47) = 0.1236 ) as ( \sin 47° ).

Key Takeaways

Frequently Asked Questions