Sin 50 Degrees : Value 0.7660 and How to Find It

What Does Sin 50 Degrees Mean?

Sine is one of the three trigonometric ratios: in a right triangle it is the side opposite the angle divided by the hypotenuse. So $\sin 50^\circ$ is the fraction of the hypotenuse taken by the opposite side when one angle is $50^\circ$.

On the unit circle, sine is the $y$-coordinate where the rotated radius meets the circle. At $50^\circ$ the radius is more than halfway up toward vertical, landing near $(0.6428, 0.7660)$, so the sine is $0.7660$.

Where Does Sin 50 Degrees Show Up?

Mid-range angles run through surveying, ramps, and structural design, where a support or a line of sight sits well off the horizontal. At $50^\circ$, $\sin 50^\circ \approx 0.766$ means the vertical component is about three-quarters of the length, which shapes the height of a leaning strut or the rise of a steep staircase.

The value also appears in the applications of trigonometry to heights and distances, where an angle of elevation near $50^\circ$ turns a measured distance into a height. It is read as the $y$-coordinate on the unit circle.

Standard-Angle Reference Table

The special first-quadrant angles have exact sine values worth memorising. Fifty degrees is not one of them, so it sits between $45^\circ$ and $60^\circ$ as a decimal only.

Since $50^\circ$ is not built from a $30$-$60$-$90$ or $45$-$45$-$90$ triangle, it has no tidy exact form; treat it as an approximation skill, not a memorisation target. The table does give a useful check: $\sin 50^\circ$ must land between $\sin 45^\circ$ and $\sin 60^\circ$, which it does. The nearby sin 47° behaves the same way.

How Do You Find The Value Of Sin 50 Degrees?

Because $50^\circ$ is not a special angle, you approximate it rather than derive a radical. Here are three routes.

Method 1: Calculator (degree mode).

Set the calculator to degrees and enter $\sin(50)$, which returns $0.76604444\ldots$. Rounded, $\sin 50^\circ \approx 0.7660$.

Method 2: The cofunction link.

Sine and cosine are cofunctions: $\sin\theta = \cos(90^\circ - \theta)$. So $\sin 50^\circ = \cos 40^\circ$, a fact from the cofunction identities that lets a cosine table return the same value.

Method 3: Bracket between known angles.

The reference table shows $\sin 45^\circ = 0.7071$ and $\sin 60^\circ = 0.8660$. Because $50^\circ$ sits between them and closer to $45^\circ$, the value must land a little above $0.7071$; the calculator confirms $0.7660$. Note that the small-angle shortcut $\sin\theta \approx \theta$ does not apply here: in radians, $50^\circ = \frac{5\pi}{18} \approx 0.8727$, which is far from the true $0.7660$, because the shortcut only holds for angles under roughly $10^\circ$ to $15^\circ$.

Examples Of Sin 50 Degrees

Example 1

Evaluate $10\sin 50^\circ$, rounded to three decimals.

$$10\sin 50^\circ = 10 \times 0.766044 = 7.66044 \approx 7.660$$

Example 2

Estimate $\sin 50^\circ$ with the small-angle rule. A student writes $\sin 50^\circ \approx 0.8727$.

Wrong attempt. The student converts to radians, $50^\circ = \frac{5\pi}{18} \approx 0.8727$, then uses $\sin\theta \approx \theta$ to get $0.8727$.

That overshoots badly: the true value is $0.7660$, an error of more than $14$. The shortcut is only valid for small angles, and $50^\circ$ is not small.

Correct. For a mid-range angle, use the calculator or the cofunction: $\sin 50^\circ = \cos 40^\circ \approx 0.7660$. Reserve $\sin\theta \approx \theta$ for angles under about $15^\circ$.

Example 3

Find $\dfrac{\sin 50^\circ}{\cos 40^\circ}$.

Because $\sin 50^\circ = \cos 40^\circ$, the ratio is:

$$\frac{\sin 50^\circ}{\cos 40^\circ} = \frac{0.766044}{0.766044} = 1$$

Example 4

A ladder $6$ m long leans against a wall at $50^\circ$ to the ground. Find how high up the wall it reaches.

$$\text{height} = 6 \times \sin 50^\circ = 6 \times 0.766044 \approx 4.596 \text{ m}$$

Example 5

Verify that $\sin 50^\circ = \sin 130^\circ$ using the supplement rule $\sin(180^\circ - \theta) = \sin\theta$.

Since $130^\circ = 180^\circ - 50^\circ$, the rule gives $\sin 130^\circ = \sin 50^\circ \approx 0.7660$. Supplementary angles share the same sine, which is why a second-quadrant $130^\circ$ matches this first-quadrant value.

Where Students Trip Up On Sin 50 Degrees

Mistake 1: Misusing the small-angle approximation

Where it slips in: Applying $\sin\theta \approx \theta$ (radians) to an angle that is not small.

Don't do this: Writing $\sin 50^\circ \approx 0.8727$ from the radian measure.

The correct way: Use the calculator or $\cos 40^\circ$ to get $0.7660$. The learner who applies the small-angle rule everywhere is the one who overshoots on mid-range angles; keep it below about $15^\circ$.

Mistake 2: Expecting a clean radical form

Where it slips in: Assuming every angle simplifies to something like $\frac{\sqrt{3}}{2}$.

Don't do this: Reporting a made-up "simplified" surd for $\sin 50^\circ$.

The correct way: State $\sin 50^\circ \approx 0.7660$ as a decimal, or leave it as $\sin 50^\circ$. Only special-triangle angles have tidy exact forms.

Mistake 3: Confusing sin 50° with sin 5°

Where it slips in: Misreading the angle and dropping or adding a zero.

Don't do this: Writing $\sin 50^\circ \approx 0.0872$, which is actually $\sin 5^\circ$.

The correct way: Check the size: $50^\circ$ is well up the first quadrant, so its sine is large ($0.7660$), while sin 5° is tiny ($0.0872$). A quick bracket against $\sin 45^\circ$ catches this.

Key Takeaways

• Sin 50 degrees is about $0.7660$; it has no simple radical form because $50^\circ$ is not a special angle.

• In radians $50^\circ = \frac{5\pi}{18}$, and the value must sit between $\sin 45^\circ$ and $\sin 60^\circ$, which it does.

• The small-angle approximation fails here; it is reliable only below roughly $10^\circ$ to $15^\circ$.

• By the cofunction rule $\sin 50^\circ = \cos 40^\circ$, and by the supplement rule $\sin 50^\circ = \sin 130^\circ$.

To build confidence with angles of elevation and estimation, explore Bhanzu's trigonometry tutor or high school math tutor, or browse math classes online.

Practice These Before Moving On

1. Evaluate $4\sin 50^\circ - 1$, rounded to three decimals.

2. Use $\sin 50^\circ = \cos 40^\circ$ to rewrite $\sin 50^\circ - \cos 50^\circ$ in terms of cosines.

3. A kite string $30$ m long makes a $50^\circ$ angle of elevation. Find the kite's height using $\sin 50^\circ$.

Want a live Bhanzu trainer to walk through more sin 50 degrees problems? Book a free demo class.

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