Trigonometry Formulas

Sine, cosine, tangent and cotangent of an acute angle, the fundamental identity, sum and double-angle formulas, the laws of sines and cosines, degree–radian conversion and tables of standard values.

20 formulas · Grades 8–10

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All 20 formulas on this page
  1. Sine of an acute angle (Sin)
  2. Cosine of an acute angle (Cos)
  3. Tangent of an acute angle (Tan)
  4. Law of cosines
  5. Law of sines
  6. Fundamental trigonometric identity
  7. Relation between tangent, sine and cosine
  8. Cotangent of an acute angle
  9. Table of sine values
  10. Table of cosine values
  11. Table of tangent values
  12. Table of cotangent values
  13. Table of angles in radians
  14. Sine of a sum and a difference
  15. Cosine of a sum and a difference
  16. Sine of a double angle
  17. Cosine of a double angle
  18. Tangent of a double angle
  19. Converting degrees to radians
  20. Converting radians to degrees

Sine of an acute angle (Sin)

Grade 8
sin⁡α=ac\sin \alpha = \frac{a}{c}

Ratio of the opposite leg to the hypotenuse.

αac
a is the opposite leg, c is the hypotenuse.
Legend:
  • opp — Opposite leg
  • hyp — Hypotenuse

Cosine of an acute angle (Cos)

Grade 8
cos⁡α=bc\cos \alpha = \frac{b}{c}

Ratio of the adjacent leg to the hypotenuse.

αbc
b is the adjacent leg, c is the hypotenuse.
Legend:
  • adj — Adjacent leg
  • hyp — Hypotenuse

Tangent of an acute angle (Tan)

Grade 8
tg⁡α=ab\operatorname{tg} \alpha = \frac{a}{b}

Ratio of the opposite leg to the adjacent leg.

αab
a is the opposite leg, b is the adjacent leg.
Legend:
  • opp — Opposite leg
  • adj — Adjacent leg

Law of cosines

Grade 9
a2=b2+c2−2bccos⁡αa^2 = b^2 + c^2 - 2bc \cos \alpha

Finds side a from two known sides and the angle between them.

αcba
a is the side opposite angle α.
Legend:
  • b — Side b
  • c — Side c
  • deg — Angle α (in degrees)

Law of sines

Grade 9
asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Calculates the circumradius R of a triangle from side a and angle A.

R
R is the radius of the circle circumscribed around the triangle.
Legend:
  • a — Side a
  • degA — Angle A (in degrees)

Fundamental trigonometric identity

Grade 9
sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1

Calculates |sin α| from a known cos α.

sin αcos α1
Follows from the Pythagorean theorem in the unit circle.
Legend:
  • cos — Cosine (cos α)

Relation between tangent, sine and cosine

Grade 9
tg⁡α=sin⁡αcos⁡α\operatorname{tg} \alpha = \frac{\sin \alpha}{\cos \alpha}

Calculating tan α as the ratio of sin α to cos α.

αsin αcos α
Tangent is the ratio of sine to cosine of the same angle.
Legend:
  • sin — Sine (sin α)
  • cos — Cosine (cos α)

Cotangent of an acute angle

Grade 9
ctg⁡ α=ba\operatorname{ctg}\,\alpha = \frac{b}{a}

Ratio of the adjacent leg to the opposite leg.

αab
Cotangent is the adjacent leg divided by the opposite leg.
Legend:
  • adj — Adjacent leg
  • opp — Opposite leg

Table of sine values

Grade 9

Values of sin α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
sin α01/2√2/2√3/21√3/20-10

Table of cosine values

Grade 9

Values of cos α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
cos α1√3/2√2/21/20-1/2-101

Table of tangent values

Grade 9

Values of tan α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
tg α01/√31√3—-√30—0

Table of cotangent values

Grade 9

Values of cot α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
ctg α—√311/√30-1/√3—0—

Table of angles in radians

Grade 9

Correspondence between standard angles in degrees and radians.

α0°30°45°60°90°120°180°270°360°
α (рад)0π/6π/4π/3π/22π/3π3π/22π

Sine of a sum and a difference

Grade 10
sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta

Calculates sin(α + β) and sin(α − β).

Legend:
  • degA — Angle α (deg)
  • degB — Angle β (deg)

Cosine of a sum and a difference

Grade 10
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta

Calculates cos(α + β) and cos(α − β).

Legend:
  • degA — Angle α (deg)
  • degB — Angle β (deg)

Sine of a double angle

Grade 10
sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha = 2 \sin \alpha \cos \alpha

Formula for expanding the sine of a double angle.

Legend:
  • deg — Angle α (in degrees)

Cosine of a double angle

Grade 10
cos⁡2α=cos⁡2α−sin⁡2α\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha

Formula for expanding the cosine of a double angle.

Legend:
  • deg — Angle α (in degrees)

Tangent of a double angle

Grade 10
tg⁡2α=2tg⁡α1−tg⁡2α\operatorname{tg} 2\alpha = \frac{2 \operatorname{tg} \alpha}{1 - \operatorname{tg}^2 \alpha}

Formula for the tangent of a double angle.

Legend:
  • deg — Angle α (in degrees)

Converting degrees to radians

Grade 10
rad=α⋅π180\mathrm{rad} = \alpha \cdot \frac{\pi}{180}

Converts an angle from degrees to radians.

α°рад
180° corresponds to π radians.
Legend:
  • alpha — Angle α (deg)

Converting radians to degrees

Grade 10
α=rad⋅180π\alpha = \mathrm{rad} \cdot \frac{180}{\pi}

Converts an angle from radians to degrees.

рад°
The reverse conversion: from radians to degrees.
Legend:
  • rad — Angle in radians (rad)

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