Find Sine, Cosine and Tangent from Sides or an Angle

Name the reference angle first

The same right triangle has two acute angles, and each uses a different opposite leg. Mark the angle θ before deciding which side is opposite and which is adjacent. The hypotenuse is always opposite the right angle and remains the longest side. The adjacent leg touches θ but is not the hypotenuse. A drawing does not need to be to scale, but its labels must show which angle and lengths belong together. Without that reference, a correct fraction can answer the wrong angle's question.

Calculate all three ratios from a worked triangle

Take a right triangle with opposite leg 8, adjacent leg 15 and hypotenuse 17 relative to θ. The three ratios are sinθ=8/17≈0.47058824, cosθ=15/17≈0.88235294 and tanθ=8/15≈0.53333333. The lengths satisfy 8²+15²=17², providing a check that they form a right triangle. These ratios have no length unit because matching units cancel. If one side is supplied in metres and another in centimetres, convert them to a common unit before dividing.

Change the angle and see which ratios change

For the other acute angle in the same 8–15–17 triangle, the opposite and adjacent legs exchange roles. Its sine is 15/17, cosine is 8/17 and tangent is 15/8=1.875. This angle is the complement of θ, so the sine of one equals the cosine of the other. The two tangent values are reciprocals because both legs are nonzero. This comparison is a useful way to diagnose swapped side labels without assuming the original arithmetic was incorrect.

Evaluate a known angle in the form

For 30 degrees, select Degrees and enter 30. The local tool returns sine 0.5, cosine approximately 0.8660254 and tangent approximately 0.57735027. Changing the input to 45 degrees gives equal sine and cosine values near 0.70710678 and tangent 1. These are function values, not side lengths. To find the opposite side of a right triangle with hypotenuse 20 at 30 degrees, multiply 20 by 0.5 to get 10; the calculator's sine value alone is not the side measurement.

Extend beyond acute triangle angles

For 150 degrees, the unit-circle point lies in the second quadrant. Sine remains positive, cosine is negative and tangent is negative. Their values are 0.5,−√3/2 and −1/√3. A right-triangle side length is positive, so the negative signs here come from directed unit-circle coordinates rather than negative physical lengths. The reference angle is 30 degrees, which supplies the magnitudes, while the quadrant supplies the signs. Do not apply the all-positive acute-triangle picture to every rotation.

Use identities as numerical cross-checks

For any real angle, sin²θ+cos²θ=1. In the 8–15–17 example, the squared ratios sum to 64/289+225/289=1 exactly. Where cosine is nonzero, tanθ=sinθ/cosθ; dividing 8/17 by 15/17 recovers 8/15. Rounded decimal values may give a result slightly different from one or from the separately rounded tangent. Retain the exact fractions or extra digits while checking, and distinguish a rounding difference from a wrong side or sign.

Recognize where a ratio is undefined

At 90 degrees, cosine is zero and sine is one, so tangent would require division by zero and is undefined. Sine and cosine remain defined. Near 90 degrees, tangent can be very large and sensitive to tiny changes in the input angle. Do not round a near-right angle to 90 before computing if that distinction matters. Similarly, the reciprocal 1/sinθ is undefined when sine is zero, but that does not make the sine value itself undefined.

Use an inverse function only to recover an angle

If the ratio opposite/hypotenuse is 0.6, an acute angle is arcsin 0.6≈36.8698976 degrees. This reverses sine and is different from 1/0.6. The local forward-angle form does not supply an arcsin input mode, so use a calculator with the required inverse function or a suitable exact-angle relationship. For a general equation sinθ=0.6, another solution in a full turn is approximately 143.1301024 degrees. The diagram or stated interval determines which solutions are relevant.

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