Trigonometry calculator
Enter an angle, get all six trig functions.
What are the six trigonometric functions?
sin, cos and tan, and their reciprocals csc = 1/sin, sec = 1/cos and cot = 1/tan. For a right triangle, sin is opposite ÷ hypotenuse, cos is adjacent ÷ hypotenuse and tan is opposite ÷ adjacent.
What are the exact values for special angles?
sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 60° = 1/2; tan 30° = √3/3, tan 45° = 1, tan 60° = √3. The calculator shows exact values for every multiple of 30° and 45°.
Why is tan 90° undefined?
tan = sin ÷ cos, and cos 90° = 0, so the division has no value. The same happens for sec 90°, and for csc and cot at 0° and 180°.
How do the signs change around the circle?
In the second quadrant (90°–180°) only sin and csc are positive, in the third (180°–270°) only tan and cot, and in the fourth (270°–360°) only cos and sec.
Can I enter radians?
Yes — switch the unit. π/6 rad (0.5236) is 30°; the calculator shows both.
How do I find the functions of an angle above 90° or a negative angle?
Find the reference angle, the acute angle between the terminal side and the horizontal axis, take the values for it, and set the signs by quadrant (OpenStax Precalculus 2e, section 5.2). For 150° the reference angle is 30° in the second quadrant, so sin 150° = 1/2 and cos 150° = −√3/2.
Why does sin 30 not give 0.5?
The unit is set to radians. Thirty radians is nearly five full turns, and sin(30 rad) = −0.9880. Switch to degrees, or enter π/6 = 0.5236 rad, to get sin 30° = 1/2.
How can I check a result?
On the unit circle cos²t + sin²t = 1 for every angle (OpenStax Precalculus 2e, section 5.2). For 5π/7 radians the calculator gives sin = 0.7818 and cos = −0.6235, and the squares add up to 1.0000.
How the values are calculated
After a turn of t from the positive x-axis, the point on the unit circle is (cos t, sin t). The other four functions are ratios of those coordinates: tan t = sin t ÷ cos t, sec t = 1 ÷ cos t, csc t = 1 ÷ sin t and cot t = cos t ÷ sin t (OpenStax Precalculus 2e, sections 5.2 and 5.3). A ratio is undefined where its denominator is 0: tan and sec at 90° and 270°, csc and cot at 0° and 180°. The calculator converts radians to degrees, and for multiples of 30° or 45° also gives the exact form from the reference angle and quadrant signs.
Worked example: −5π/6
OpenStax Example 3 asks for the six functions of −5π/6, which is −150°. The reference angle is 30° and the terminal side is in the third quadrant. Entering −150 degrees gives sin = −1/2, cos = −√3/2, tan = √3/3, csc = −2, sec = −2√3/3 and cot = √3, the answers in the book. For an angle without an exact form, such as 5π/7 (about 128.57°), the calculator shows decimals: csc = 1.279, which is the 1.279 from Example 10.
Common mistakes
Mixing up degree and radian mode causes most wrong answers; see the question above. Csc is not the inverse function of sine: csc is 1 ÷ sin, while the inverse returns an angle from a ratio, which this tool does not do. Near an undefined point values explode rather than fail: tan 89.9° is 572.96, tan 90° is undefined.
Sources: OpenStax, Precalculus 2e, section 5.2 “Unit Circle: Sine and Cosine Functions” and section 5.3 “The Other Trigonometric Functions” (definitions, Table 1, Examples 3 and 10). Related tools: right triangle calculator, triangle calculator, degrees to radians converter and Pythagorean theorem calculator. How we check our calculators.