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Important Formulas & Examples: Trigonometric Functions - 1
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Page 1 Trigonometric Formula Sheet De?nition of the Trig Functions Right Triangle De?nition Assume that: 0 < ? < ? 2 or 0 < ? < 90 hypotenuse adjacent opposite ? sin? = opp hyp csc? = hyp opp cos? = adj hyp sec? = hyp adj tan? = opp adj cot? = adj opp Unit Circle De?nition Assume ? can be any angle. x y y x 1 (x,y) ? sin? = y 1 csc? = 1 y cos? = x 1 sec? = 1 x tan? = y x cot? = x y Domains of the Trig Functions sin? , 8 ? 2 ( 1 ,1 ) cos? , 8 ? 2 ( 1 ,1 ) tan? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z csc? , 8 ? 6= n? ,wheren2 Z sec? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z cot? , 8 ? 6= n? ,wheren2 Z Ranges of the Trig Functions 1? sin? ? 1 1? cos? ? 1 1? tan? ?1 csc? 1 and csc? ? 1 sec? 1 and sec? ? 1 1? cot? ?1 Periods of the Trig Functions The period of a function is the number, T, such that f (? +T ) = f (? ). So, if ! is a ?xed number and ? is any angle we have the following periods. sin(!? ) ) T = 2? ! cos(!? ) ) T = 2? ! tan(!? ) ) T = ? ! csc(!? ) ) T = 2? ! sec(!? ) ) T = 2? ! cot(!? ) ) T = ? ! 1 Page 2 Trigonometric Formula Sheet De?nition of the Trig Functions Right Triangle De?nition Assume that: 0 < ? < ? 2 or 0 < ? < 90 hypotenuse adjacent opposite ? sin? = opp hyp csc? = hyp opp cos? = adj hyp sec? = hyp adj tan? = opp adj cot? = adj opp Unit Circle De?nition Assume ? can be any angle. x y y x 1 (x,y) ? sin? = y 1 csc? = 1 y cos? = x 1 sec? = 1 x tan? = y x cot? = x y Domains of the Trig Functions sin? , 8 ? 2 ( 1 ,1 ) cos? , 8 ? 2 ( 1 ,1 ) tan? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z csc? , 8 ? 6= n? ,wheren2 Z sec? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z cot? , 8 ? 6= n? ,wheren2 Z Ranges of the Trig Functions 1? sin? ? 1 1? cos? ? 1 1? tan? ?1 csc? 1 and csc? ? 1 sec? 1 and sec? ? 1 1? cot? ?1 Periods of the Trig Functions The period of a function is the number, T, such that f (? +T ) = f (? ). So, if ! is a ?xed number and ? is any angle we have the following periods. sin(!? ) ) T = 2? ! cos(!? ) ) T = 2? ! tan(!? ) ) T = ? ! csc(!? ) ) T = 2? ! sec(!? ) ) T = 2? ! cot(!? ) ) T = ? ! 1 Page 3 Trigonometric Formula Sheet De?nition of the Trig Functions Right Triangle De?nition Assume that: 0 < ? < ? 2 or 0 < ? < 90 hypotenuse adjacent opposite ? sin? = opp hyp csc? = hyp opp cos? = adj hyp sec? = hyp adj tan? = opp adj cot? = adj opp Unit Circle De?nition Assume ? can be any angle. x y y x 1 (x,y) ? sin? = y 1 csc? = 1 y cos? = x 1 sec? = 1 x tan? = y x cot? = x y Domains of the Trig Functions sin? , 8 ? 2 ( 1 ,1 ) cos? , 8 ? 2 ( 1 ,1 ) tan? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z csc? , 8 ? 6= n? ,wheren2 Z sec? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z cot? , 8 ? 6= n? ,wheren2 Z Ranges of the Trig Functions 1? sin? ? 1 1? cos? ? 1 1? tan? ?1 csc? 1 and csc? ? 1 sec? 1 and sec? ? 1 1? cot? ?1 Periods of the Trig Functions The period of a function is the number, T, such that f (? +T ) = f (? ). So, if ! is a ?xed number and ? is any angle we have the following periods. sin(!? ) ) T = 2? ! cos(!? ) ) T = 2? ! tan(!? ) ) T = ? ! csc(!? ) ) T = 2? ! sec(!? ) ) T = 2? ! cot(!? ) ) T = ? ! 1 Identities and Formulas Tangent and Cotangent Identities tan? = sin? cos? cot? = cos? sin? Reciprocal Identities sin? = 1 csc? csc? = 1 sin? cos? = 1 sec? sec? = 1 cos? tan? = 1 cot? cot? = 1 tan? Pythagorean Identities sin 2 ? +cos 2 ? =1 tan 2 ? +1=sec 2 ? 1+cot 2 ? =csc 2 ? Even and Odd Formulas sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas If n is an integer sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas sin(2? )=2sin? cos? cos(2? )=cos 2 ? sin 2 ? =2cos 2 ? 1 =1 2sin 2 ? tan(2? )= 2tan? 1 tan 2 ? Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then: ? 180 = t x ) t = ? x 180 and x = 180 t ? Half Angle Formulas sin? =± r 1 cos(2? ) 2 cos? =± r 1+cos(2? ) 2 tan? =± s 1 cos(2? ) 1+cos(2? ) Sum and Di? erence Formulas sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )= tan? ±tan 1? tan? tan Product to Sum Formulas sin? sin = 1 2 [cos(? ) cos(? + )] cos? cos = 1 2 [cos(? )+cos(? + )] sin? cos = 1 2 [sin(? + )+sin(? )] cos? sin = 1 2 [sin(? + ) sin(? )] Sum to Product Formulas sin? +sin =2sin ? ? + 2 ? cos ? ? 2 ? sin? sin =2cos ? ? + 2 ? sin ? ? 2 ? cos? +cos =2cos ? ? + 2 ? cos ? ? 2 ? cos? cos = 2sin ? ? + 2 ? sin ? ? 2 ? Cofunction Formulas sin ? ? 2 ? ? =cos? csc ? ? 2 ? ? =sec? tan ? ? 2 ? ? =cot? cos ? ? 2 ? ? =sin? sec ? ? 2 ? ? =csc? cot ? ? 2 ? ? =tan? 2 Page 4 Trigonometric Formula Sheet De?nition of the Trig Functions Right Triangle De?nition Assume that: 0 < ? < ? 2 or 0 < ? < 90 hypotenuse adjacent opposite ? sin? = opp hyp csc? = hyp opp cos? = adj hyp sec? = hyp adj tan? = opp adj cot? = adj opp Unit Circle De?nition Assume ? can be any angle. x y y x 1 (x,y) ? sin? = y 1 csc? = 1 y cos? = x 1 sec? = 1 x tan? = y x cot? = x y Domains of the Trig Functions sin? , 8 ? 2 ( 1 ,1 ) cos? , 8 ? 2 ( 1 ,1 ) tan? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z csc? , 8 ? 6= n? ,wheren2 Z sec? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z cot? , 8 ? 6= n? ,wheren2 Z Ranges of the Trig Functions 1? sin? ? 1 1? cos? ? 1 1? tan? ?1 csc? 1 and csc? ? 1 sec? 1 and sec? ? 1 1? cot? ?1 Periods of the Trig Functions The period of a function is the number, T, such that f (? +T ) = f (? ). So, if ! is a ?xed number and ? is any angle we have the following periods. sin(!? ) ) T = 2? ! cos(!? ) ) T = 2? ! tan(!? ) ) T = ? ! csc(!? ) ) T = 2? ! sec(!? ) ) T = 2? ! cot(!? ) ) T = ? ! 1 Identities and Formulas Tangent and Cotangent Identities tan? = sin? cos? cot? = cos? sin? Reciprocal Identities sin? = 1 csc? csc? = 1 sin? cos? = 1 sec? sec? = 1 cos? tan? = 1 cot? cot? = 1 tan? Pythagorean Identities sin 2 ? +cos 2 ? =1 tan 2 ? +1=sec 2 ? 1+cot 2 ? =csc 2 ? Even and Odd Formulas sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas If n is an integer sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas sin(2? )=2sin? cos? cos(2? )=cos 2 ? sin 2 ? =2cos 2 ? 1 =1 2sin 2 ? tan(2? )= 2tan? 1 tan 2 ? Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then: ? 180 = t x ) t = ? x 180 and x = 180 t ? Half Angle Formulas sin? =± r 1 cos(2? ) 2 cos? =± r 1+cos(2? ) 2 tan? =± s 1 cos(2? ) 1+cos(2? ) Sum and Di? erence Formulas sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )= tan? ±tan 1? tan? tan Product to Sum Formulas sin? sin = 1 2 [cos(? ) cos(? + )] cos? cos = 1 2 [cos(? )+cos(? + )] sin? cos = 1 2 [sin(? + )+sin(? )] cos? sin = 1 2 [sin(? + ) sin(? )] Sum to Product Formulas sin? +sin =2sin ? ? + 2 ? cos ? ? 2 ? sin? sin =2cos ? ? + 2 ? sin ? ? 2 ? cos? +cos =2cos ? ? + 2 ? cos ? ? 2 ? cos? cos = 2sin ? ? + 2 ? sin ? ? 2 ? Cofunction Formulas sin ? ? 2 ? ? =cos? csc ? ? 2 ? ? =sec? tan ? ? 2 ? ? =cot? cos ? ? 2 ? ? =sin? sec ? ? 2 ? ? =csc? cot ? ? 2 ? ? =tan? 2 Unit Circle 0 ,2? (1,0) 180 ,? ( 1,0) (0,1) 90 , ? 2 (0, 1) 270 , 3? 2 30 , ? 6 ( p 3 2 , 1 2 ) 45 , ? 4 ( p 2 2 , p 2 2 ) 60 , ? 3 ( 1 2 , p 3 2 ) 120 , 2? 3 ( 1 2 , p 3 2 ) 135 , 3? 4 ( p 2 2 , p 2 2 ) 150 , 5? 6 ( p 3 2 , 1 2 ) 210 , 7? 6 ( p 3 2 , 1 2 ) 225 , 5? 4 ( p 2 2 , p 2 2 ) 240 , 4? 3 ( 1 2 , p 3 2 ) 300 , 5? 3 ( 1 2 , p 3 2 ) 315 , 7? 4 ( p 2 2 , p 2 2 ) 330 , 11? 6 ( p 3 2 , 1 2 ) For any ordered pair on the unit circle (x,y):cos? = x and sin? = y Example cos( 7? 6 )= p 3 2 sin( 7? 6 )= 1 2 3 Page 5 Trigonometric Formula Sheet De?nition of the Trig Functions Right Triangle De?nition Assume that: 0 < ? < ? 2 or 0 < ? < 90 hypotenuse adjacent opposite ? sin? = opp hyp csc? = hyp opp cos? = adj hyp sec? = hyp adj tan? = opp adj cot? = adj opp Unit Circle De?nition Assume ? can be any angle. x y y x 1 (x,y) ? sin? = y 1 csc? = 1 y cos? = x 1 sec? = 1 x tan? = y x cot? = x y Domains of the Trig Functions sin? , 8 ? 2 ( 1 ,1 ) cos? , 8 ? 2 ( 1 ,1 ) tan? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z csc? , 8 ? 6= n? ,wheren2 Z sec? , 8 ? 6= ? n+ 1 2 ? ? ,wheren2 Z cot? , 8 ? 6= n? ,wheren2 Z Ranges of the Trig Functions 1? sin? ? 1 1? cos? ? 1 1? tan? ?1 csc? 1 and csc? ? 1 sec? 1 and sec? ? 1 1? cot? ?1 Periods of the Trig Functions The period of a function is the number, T, such that f (? +T ) = f (? ). So, if ! is a ?xed number and ? is any angle we have the following periods. sin(!? ) ) T = 2? ! cos(!? ) ) T = 2? ! tan(!? ) ) T = ? ! csc(!? ) ) T = 2? ! sec(!? ) ) T = 2? ! cot(!? ) ) T = ? ! 1 Identities and Formulas Tangent and Cotangent Identities tan? = sin? cos? cot? = cos? sin? Reciprocal Identities sin? = 1 csc? csc? = 1 sin? cos? = 1 sec? sec? = 1 cos? tan? = 1 cot? cot? = 1 tan? Pythagorean Identities sin 2 ? +cos 2 ? =1 tan 2 ? +1=sec 2 ? 1+cot 2 ? =csc 2 ? Even and Odd Formulas sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas If n is an integer sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas sin(2? )=2sin? cos? cos(2? )=cos 2 ? sin 2 ? =2cos 2 ? 1 =1 2sin 2 ? tan(2? )= 2tan? 1 tan 2 ? Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then: ? 180 = t x ) t = ? x 180 and x = 180 t ? Half Angle Formulas sin? =± r 1 cos(2? ) 2 cos? =± r 1+cos(2? ) 2 tan? =± s 1 cos(2? ) 1+cos(2? ) Sum and Di? erence Formulas sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )= tan? ±tan 1? tan? tan Product to Sum Formulas sin? sin = 1 2 [cos(? ) cos(? + )] cos? cos = 1 2 [cos(? )+cos(? + )] sin? cos = 1 2 [sin(? + )+sin(? )] cos? sin = 1 2 [sin(? + ) sin(? )] Sum to Product Formulas sin? +sin =2sin ? ? + 2 ? cos ? ? 2 ? sin? sin =2cos ? ? + 2 ? sin ? ? 2 ? cos? +cos =2cos ? ? + 2 ? cos ? ? 2 ? cos? cos = 2sin ? ? + 2 ? sin ? ? 2 ? Cofunction Formulas sin ? ? 2 ? ? =cos? csc ? ? 2 ? ? =sec? tan ? ? 2 ? ? =cot? cos ? ? 2 ? ? =sin? sec ? ? 2 ? ? =csc? cot ? ? 2 ? ? =tan? 2 Unit Circle 0 ,2? (1,0) 180 ,? ( 1,0) (0,1) 90 , ? 2 (0, 1) 270 , 3? 2 30 , ? 6 ( p 3 2 , 1 2 ) 45 , ? 4 ( p 2 2 , p 2 2 ) 60 , ? 3 ( 1 2 , p 3 2 ) 120 , 2? 3 ( 1 2 , p 3 2 ) 135 , 3? 4 ( p 2 2 , p 2 2 ) 150 , 5? 6 ( p 3 2 , 1 2 ) 210 , 7? 6 ( p 3 2 , 1 2 ) 225 , 5? 4 ( p 2 2 , p 2 2 ) 240 , 4? 3 ( 1 2 , p 3 2 ) 300 , 5? 3 ( 1 2 , p 3 2 ) 315 , 7? 4 ( p 2 2 , p 2 2 ) 330 , 11? 6 ( p 3 2 , 1 2 ) For any ordered pair on the unit circle (x,y):cos? = x and sin? = y Example cos( 7? 6 )= p 3 2 sin( 7? 6 )= 1 2 3Read More
FAQs on Important Formulas & Examples: Trigonometric Functions - 1
| 1. What are the six trigonometric ratios and how do I remember them easily for CBSE exams? |  |
Ans. The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. Students often use the mnemonic SOH-CAH-TOA to remember sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent). The reciprocal functions-cosecant, secant, and cotangent-are simply 1/sin, 1/cos, and 1/tan respectively. Flashcards and mind maps help reinforce these relationships quickly before exams.
| 2. Why do trigonometric functions have different values at 0°, 30°, 45°, 60°, and 90°? | |
Ans. These special angles produce exact trigonometric values based on specific right-angled triangle ratios. At 0°, sine equals 0 and cosine equals 1; at 90°, sine equals 1 and cosine equals 0. The 30-60-90 and 45-45-90 triangles generate predictable ratios like √3/2 and 1/√2 that appear repeatedly in problems. Understanding this pattern prevents calculation errors in Grade 12 mathematics.
| 3. How do I know which trigonometric function to use when solving a right triangle problem? | |
Ans. Identify which sides you know and which you need to find. If you have the opposite side and hypotenuse, use sine; for adjacent and hypotenuse, use cosine; for opposite and adjacent, use tangent. This selection method-matching available information to the correct trigonometric ratio-is essential for Grade 12 problem-solving and ensures you don't waste time trying wrong approaches during board exams.
| 4. What's the difference between sin²θ + cos²θ = 1 and other trigonometric identities? | |
Ans. The Pythagorean identity sin²θ + cos²θ = 1 is fundamental-it applies to all angles and forms the basis for deriving other identities like 1 + tan²θ = sec²θ. Other identities are specific to angle relationships, reciprocals, or sum-difference formulas. Mastering this core identity unlocks simplification of complex trigonometric expressions, which frequently appears in CBSE Class 12 examinations.
| 5. Can trigonometric function values be negative, and when does this happen? | |
Ans. Yes, trigonometric function values are negative in specific quadrants. Sine is negative in the third and fourth quadrants; cosine is negative in the second and third quadrants; tangent is negative in the second and fourth quadrants. The ASTC rule (All-Students-Take-Calculus) helps track this. Understanding sign changes across quadrants prevents critical mistakes in evaluating trigonometric expressions and angle-based calculations.
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