Trigonometry Formulas: Ratios, Identities and Triangles

Match the formula to the information supplied

A useful trigonometry formula sheet should distinguish a ratio definition, an identity and a triangle-solving theorem. A ratio evaluates a function from an angle or side pair. An identity relates expressions for every angle where they are defined. A triangle theorem connects multiple sides and angles. Start by identifying whether a triangle is right angled and which quantities are known. The label A Math alone does not identify an examination board or guarantee that every formula here belongs to its current syllabus.

Use the three right-triangle ratios

Relative to an acute angle θ, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. The hypotenuse lies opposite the right angle; the opposite and adjacent legs change roles when you choose the other acute angle. In a triangle with legs 9 and 12 and hypotenuse 15, the angle opposite the side 9 has sine 9/15=0.6, cosine 12/15=0.8 and tangent 9/12=0.75. Its complement swaps the sine and cosine values, so label the angle before substituting.

Rearrange the chosen ratio before evaluating

Suppose a right triangle has hypotenuse 14 centimetres and angle 30 degrees, with its opposite side unknown. From sin 30°=opposite/14, the unknown side is 14 sin 30°=7 centimetres. The local form supplies sin 30°=0.5, then you multiply by 14 separately. Dividing 14 by sin 30° would give 28, larger than the known hypotenuse, and signals an incorrect rearrangement. A side-length result should satisfy both the algebraic equation and the geometric size constraint.

Apply identities with their domain restrictions

The Pythagorean identity sin²θ+cos²θ=1 links the two unit-circle coordinates. The quotient identity tanθ=sinθ/cosθ requires nonzero cosine. At 60 degrees, sinθ=√3/2 and cosθ=1/2, so their squared values sum to 3/4+1/4=1 and their quotient is √3. A decimal calculation can be slightly different from an exact identity because of rounding. Do not divide by a zero cosine or erase a denominator without retaining the original restriction on the angle.

Recognize sum-angle and double-angle formulas

The identities sin(A+B)=sin A cos B+cos A sin B and cos(A+B)=cos A cos B−sin A sin B lead to sin(2A)=2 sin A cos A and cos(2A)=cos²A−sin²A. For A=30 degrees, the double-angle sine is 2(1/2)(√3/2)=√3/2, agreeing with sin 60 degrees. The angle sum is inside the function: sin(A+B) generally does not equal sin A+sin B. Write the factors explicitly when checking a symbolic identity, and use consistent angle units throughout the substitution.

Use a suitable theorem for a non-right triangle

For sides a,b,c opposite angles A,B,C, the sine rule is a/sinA=b/sinB=c/sinC. The cosine rule c²=a²+b²−2 ab cos C is useful when two sides and their included angle are known. If a=7, b=10 and C=60 degrees, c²=49+100−140(1/2)=79, so c=√79≈8.8882. The triangle area is ½ab sin C=35 sin 60°≈30.3109 square units. These calculations use several quantities; entering only 60 in the local trig form supplies function values, not the completed triangle.

Keep inverse functions and reciprocals separate

The inverse function arcsin reverses sine within a chosen principal range, whereas cosecant is the reciprocal 1/sinθ. The notation sin^−1 can be confusing because it commonly denotes arcsin in calculator contexts rather than the reciprocal. For a right-triangle ratio 0.6, arcsin gives about 36.8699 degrees; the reciprocal of 0.6 is 1.6667, which is not that angle. Solving a trigonometric equation can require additional angles beyond an inverse function's principal answer.

Check the result against its context

Sine and cosine must lie between −1 and 1, but tangent need not. An acute right-triangle angle has positive ratios, while unit-circle signs depend on the quadrant. Check whether the task expects exact radicals, degrees, radians or a specified decimal precision before reporting the answer. Use 30 degrees and 60 degrees as independent local checks: their sine and cosine values exchange, and their tangents are approximately 0.57735027 and 1.73205081. A different pattern often points to the wrong unit setting or a swapped ratio.

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