1. Trigonometric function of the same angle | sin2Ώ{cos2Ώ1 | sin Ώ * cosec Ώ1 |
---|---|---|
sec2Ώ|tan2Ώ1 | cos Ώ * sec Ώ1 | |
cosec2Ώ|cot2Ώ1 | tan Ώ * cot Ώ1 | |
sin Ώ / cos Ώtan Ώ | cos Ώ / sin Ώcot Ώ | |
2. Representing one trigonometric function with another trigonometric function (of the same angle) (The radicals of these formulas must be preceded by a "+" or "-" sign depending on which quadrant angle Ώ is.) |
sin Ώ (1|cos2Ώ)0.5 @@@ tan Ώ /(1{tan2Ώ)0.5 @@@ 1 / (1{cot2Ώ)0.5 @@@ (sec2Ώ|1)0.5 / sec Ώ @@@ 1 / cosec Ώ |
tan Ώ sin Ώ / (1|sin2Ώ)0.5 @@@ (1|cos2Ώ)0.5 / cos Ώ @@@ 1 / cot Ώ @@@ (sec2Ώ|1)0.5 @@@ 1 / (cosec2Ώ|1)0.5 |
cos Ώ (1|sin2Ώ)0.5 @@@ 1 /(1{tan2Ώ)0.5 @@@ cot Ώ / (1{cot2Ώ)0.5 @@@ 1 / sec Ώ @@@ (cosec2Ώ|1)0.5 / cosec Ώ |
cot Ώ (1|sin2Ώ)0.5 / sin Ώ @@@ cos Ώ / (1|cos2Ώ)0.5 @@@ 1 / tan Ώ @@@ 1 / (sec2Ώ|1)0.5 @@@ (cosec2Ώ|1)0.5 | |
3. Trigonometric function of sum and difference of angles (addition theorem) | sin (Ώ}ΐ) sin Ώ cos ΐ } cos Ώ sin ΐ | |
cos (Ώ{ΐ) cos Ώ cos ΐ | sin Ώ sin ΐ | ||
cos (Ώ|ΐ) cos Ώ cos ΐ { sin Ώ sin ΐ | ||
tan (Ώ{ΐ) (tan Ώ { tan ΐ) / (1 | tan Ώ tan ΐ) | ||
tan (Ώ|ΐ) (tan Ώ | tan ΐ) / (1 { tan Ώ tan ΐ) | ||
cot (Ώ{ΐ) [(cot Ώ cot ΐ) | 1] / (cot ΐ { cot Ώ) | ||
cot (Ώ|ΐ) [(cot Ώ cot ΐ) { 1] / (cot ΐ | cot Ώ) | ||
sin (Ώ{ΐ{Α) sin Ώ cos ΐ cos Α { cos Ώ sin ΐ cos Α { cos Ώ cos ΐ sin Α | sin Ώ sin ΐ sin Α | ||
cos (Ώ{ΐ{Α) cos Ώ cos ΐ cos Α | sin Ώ sin ΐ cos Α | sin Ώ cos ΐ sin Α | cos Ώ sin ΐ sin Α | ||
4. Double angle trigonometric function | sin 2Ώ 2sin Ώ cos Ώ | cos 2Ώ cos2 Ώ | sin2Ώ |
sin 3Ώ 3sin Ώ | 4sin3Ώ | cos 3Ώ 4cos3 Ώ | 3cos Ώ | |
sin 4Ώ 8cos3 Ώ sin Ώ | 4cos Ώ sin Ώ | cos 4Ώ 8cos4 Ώ | 8cos2 Ώ { 1 | |
tan 2Ώ 2tan Ώ /(1|tan2 Ώ) | ||
tan 3Ώ (3tan Ώ | tan3 Ώ) / (1|3tan2 Ώ) | ||
tan 4Ώ (4tan Ώ | 4tan3 Ώ) / (1|6tan2 Ώ { tan4 Ώ) | ||
cot 2Ώ (cot2 Ώ | 1) /(2cot Ώ) | ||
cot 3Ώ (cot3 Ώ | 3cot Ώ) / (3cot2 Ώ | 1) | ||
cot 4Ώ (cot4 Ώ | 6cot2 Ώ { 1) / (4cot3 Ώ | 4cot Ώ) | ||
5.Half-width trigonometric function | sin (Ώ/2) [(1 | cos Ώ) / 2]0.5 | |
cos (Ώ/2) [(1 { cos Ώ) / 2]0.5 | ||
tan (Ώ/2) [(1 | cos Ώ) / (1{cos Ώ)]0.5 (1 | cos Ώ) / sin Ώ sin Ώ / (1 { cos Ώ) | ||
cot (Ώ/2) [(1 { cos Ώ) / (1|cos Ώ)]0.5 (1 { cos Ώ) / sin Ώ sin Ώ / (1 | cos Ώ) | ||
6. Sum and difference of trigonometric functions | sin Ώ { sin ΐ 2sin[(Ώ { ΐ)/2] * cos[(Ώ|ΐ)/2] | |
sin Ώ | sin ΐ 2cos[(Ώ { ΐ)/2] * sin[(Ώ|ΐ)/2] | ||
cos Ώ { cos ΐ 2cos[(Ώ { ΐ)/2] * cos[(Ώ|ΐ)/2] | ||
cos Ώ | cos ΐ -2sin[(Ώ { ΐ)/2] * sin[(Ώ | ΐ)/2] | ||
tan Ώ } tan ΐ sin (Ώ } ΐ) / (cos Ώ cos ΐ) | ||
cot Ώ } cot ΐ }sin (Ώ } ΐ) / (sin Ώ sin ΐ) | ||
tan Ώ { cot ΐ cos (Ώ | ΐ) / (cos Ώ sin ΐ) | ||
cot Ώ | tan ΐ cos (Ώ { ΐ) / (sin Ώ cos ΐ) | ||
7. Product of trigonometric functions | sin Ώ sin ΐ [cos(Ώ | ΐ) | cos(Ώ { ΐ)] / 2 | |
cos Ώ cos ΐ [cos(Ώ | ΐ) { cos(Ώ { ΐ)] / 2 | ||
sin Ώ cos ΐ [sin(Ώ | ΐ) { sin(Ώ { ΐ)] / 2 | ||
sin Ώ sin ΐ sinΑ [sin(Ώ { ΐ | Α) { sin(ΐ { Α | Ώ) { sin(Α { Ώ | ΐ) | sin(Ώ { ΐ { Α)] / 4 | ||
sin Ώ cos ΐ cosΑ [sin(Ώ { ΐ | Α) | sin(ΐ { Α | Ώ) { sin(Α { Ώ | ΐ) | sin(Ώ { ΐ { Α)] / 4 | ||
sin Ώ sin ΐ cosΑ [-cos(Ώ { ΐ | Α) { cos(ΐ { Α | Ώ) { cos(Α { Ώ | ΐ) { cos(Ώ { ΐ { Α)] / 4 | ||
cos Ώ cos ΐ cosΑ [cos(Ώ { ΐ | Α) { cos(ΐ { Α | Ώ) { cos(Α { Ώ | ΐ) { cos(Ώ { ΐ { Α)] / 4 | ||
8. Trigonometric exponentiation | sin2Ώ (1 | cos2Ώ) / 2 | cos2Ώ (1 { cos2Ώ) / 2 |
sin3Ώ (3sinΏ | sin3Ώ) / 4 | cos3Ώ (cos3Ώ { 3cosΏ) / 4 | |
sin4Ώ (cos4Ώ | 4cos2Ώ{ 3) / 8 | cos4Ώ (cos4Ώ { 4cos2Ώ{ 3) / 8 | |
References: "Basic Mathematics Handbook" (translated by Toshio Miyamoto and Nobuyuki Matsuda) Algebra and Geometry, page 170, quoted from the basic formula of trigonometry. |