Math Buddyv1.0

Trig calculator with steps

Type a trig problem. MathBuddy finds exact values, simplifies identities and solves triangles step by step.

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What this page covers

  • Exact values of sine, cosine and tangent at standard angles, using reference angles and signs by quadrant.
  • Sum and difference formulas, such as finding sin⁡75∘\sin 75^\circ exactly.
  • Sides and angles of triangles with the sine rule, the cosine rule and right-triangle ratios.
  • Not covered here: graphs of trig functions, and derivatives or integrals of trig functions (use the derivative or integral calculator).

How to enter the problem

  • Type it or paste it. Write sin, cos and tan with the angle in brackets, and say whether the angle is in degrees or radians.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Find the exact value of sin⁡75∘\displaystyle \sin 75^\circ.

Answer

Verified

The answer is

6+24\frac{\sqrt{6} + \sqrt{2}}{4}

Explanation

  1. Split the angle
    75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ, two angles with known values.
  2. Use the sum formula
    sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B
  3. Substitute the values
    22⋅32+22⋅12=64+24\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}

Example 2

Problem

In a right triangle one angle is 60∘\displaystyle 60^\circ and the side next to it (not the hypotenuse) is 4\displaystyle 4. Find the hypotenuse h\displaystyle h.

Answer

Verified

The answer is

h=8h = 8

Explanation

  1. Choose the ratio
    The adjacent side and the hypotenuse are involved, so use cosine
    cos⁡60∘=4h\cos 60^\circ = \frac{4}{h}
  2. Use the exact value
    cos⁡60∘=12\cos 60^\circ = \frac{1}{2}, so 12=4h\frac{1}{2} = \frac{4}{h}.
  3. Solve for hh
    h=4÷12=8h = 4 \div \frac{1}{2} = 8

Example 3

Problem

A triangle has sides 7\displaystyle 7 and 8\displaystyle 8 with an angle of 60∘\displaystyle 60^\circ between them. Find the third side c\displaystyle c.

Answer

Verified

The answer is

c=57c = \sqrt{57}

Explanation

  1. Choose the rule
    Two sides and the angle between them call for the cosine rule
    c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
  2. Substitute
    c2=49+64−2⋅7⋅8⋅12=113−56=57c^2 = 49 + 64 - 2 \cdot 7 \cdot 8 \cdot \frac{1}{2} = 113 - 56 = 57
  3. Take the positive root
    c=57≈7.55c = \sqrt{57} \approx 7.55

Common mistakes

  • Getting the sign wrong in the second quadrant. Tangent is negative there, so tan⁡5π6\tan\frac{5\pi}{6} is −33-\frac{\sqrt{3}}{3}, not +33+\frac{\sqrt{3}}{3}.
  • Adding angles inside the sine: sin⁡75∘\sin 75^\circ is not sin⁡45∘+sin⁡30∘\sin 45^\circ + \sin 30^\circ.
  • Mixing degrees and radians. sin⁡30\sin 30 in radians is about −0.988-0.988, not 12\frac{1}{2}.
  • Forgetting the minus term in the cosine rule, which gives c=113c = \sqrt{113} instead of 57\sqrt{57}.

Checks, assumptions and limits

  • Check by hand: compare an exact answer with a decimal. 6+24≈0.9659\frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659, and sin⁡75∘\sin 75^\circ on a calculator in degree mode gives the same.
  • State the angle unit. Problems in radians and degrees give different results for the same number.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Yes. At standard angles the answer is kept exact, with square roots and fractions. A decimal approximation is available if you ask for it.

Yes. Write the degree sign or say degrees. Angles written with pi are read as radians.

Yes. Give the sides and angles you know. The steps say which rule is used, sine rule, cosine rule or a right-triangle ratio.

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