. CONCEPTS & FORMULAS. INTRODUCTION Radian The angle subtended at centre of a circle by an arc of length equal to the radius of the circle is radian r o = o radian r r o radian = o = 6 Positive & Negative Angles A positive angle is measured in an anticlockwise direction A negative angle is measured clockwise direction. TRIGONOMETRIC FUNCTIONS CHARACTERISTICS Si Trigonometric Functions Page of A B AB BC sin ; cos ; tan AC AC AC cosec sin AB AC sec cos BC BC cot tan AB Important Trigonometric Ratios A Sine Cosine Tangent B C 6 9 C in a AB BC Trigonometry Functions Page
Sign of Trigonometric Functions Sine positive Cos negative Tan negative Sine positive Cos positive Tan positive Sine negative Cos negative Tan positive Sine negative Cos positive Tan negative Trigonometry Identities & Formulas Basic Identities : sin θ cos θ sec θ tan θ cosec θ cot θ Compound-Angle Formulas Double-Angle Formulas Factor Formulas (Sum to Product : sin(a + B = sin A cos B + cos A sin B sin(a B = sin A cos B - cos A sin B cos(a + B = cos A cos B sin A sin B cos (A - B = cos A cos B + sin A sin B tan A tan B tan( A B tan A tan B tan A tan B tan( A B tan A tan B : : sin sin cos cos cos sin tan tan tan sin cos Page of C D C D sin C sin D sin cos C D C D sin C sin D cos sin C D C D cos C cos D cos cos C D C D cos D cos C sin sin Factor Formulas (Product to Sum : sin A cos B = sin (A + B + sin (A B cos A sin B = sin (A + B - sin (A B cos A cos B = cos ( A + B + cos (A B sin A sin B = cos (A B cos (A + B Trigonometry Functions Page
. Solve each of the following equations giving all solutions in the given interval a cos 7.7, for 6. cos 7.7 cos 7.7.6 7.7., 6.7. 7.7, (6.7 7.7 6.6,6.9 ( Answer b 7sin( 6.9, for 6. 7sin( 6.9 sin( 6.9 7 6.9.,.. 6.9,,.6 6.9 7.,6. ( Answer c sin( 67., for d 6 cos(., for 6 sin( 67. sin( 67. 67. 7.,. 9.,.., (6 9..,. Since 7. Note,7. then ( Answer 6. Given that tan. Find without the use of a calculator, the eact value of tan,given that a tan( b sin( cos( 6 cos(. cos(. 6..,.67 7.,. ( Answer Page 7 of a tan( tan tan tan tan tan ( tan tan tan 9 tan tan 9 ( Answer b sin( cos( sin cos cos sin cos cos sin sin sin cos cos sin sin cos cos 6sin sin cos tan ( Answer tan sin cos Trigonometry Functions Page 7
6 Prove the identity cos cos (cos cos cos cos cos cos sin cos ( Shown cos cos cos( cos cos sin sin (Note cos (sin cos (sin cos cos cos ( cos cos cos cos 7. Prove the identity sin sin sin sin sin( sin cos cos sin (Note sin cos cos ( sin sin cos sin ( sin sin sin sin sin sin sin sin sin sin sin ( Shown sin Note: Use Compound-Angle Formula cos(a + B = cos A cos B sin A sin B Note: Use Compound-Angle Formula sin(a + B = sin A cos B + cos A sin B. Prove the identity (cos cos cos cos cos cos cos cos cos cos cos( ( cos (cos cos cos cos ( Shown sin cos ( cos cos cos sin sin cos (sin cos.sin cos cos cos (cos cos cos cos cos Page 9 of 9. Prove the identity sin cos cos sin cos cos cos (Note cos cos cos cos Let A cos A cos A cos A cos A cos cos ( Shown Note cos cos cos cos cos cos cos sin sin cos cos sin Trigonometry Functions Page 9
. Epress sin cos in the form R sin(, where R and 9, giving the eact value of R and the value of. correct to decimal places Let sin cos Rsin( sin cos Rsin cos Rcos sin Equate the coefficien ts of cos and sin Rcos...( Rsin...( ( R ( : R cos 6 R sin 6 9 R 7 ( Answer. Epress sin cos in the form R sin(, where R > and 9, giving the value of correct to decimal places. Let sin cos Rsin( sin cos Rsin cos Rcos sin Equate the coefficien ts of cos and sin Rcos...( Rsin...( ( ( : R R cos R sin 69 R ( Answer Replace R in (: cos 7 cos 6.9 ( Answer 7 in the form R cos 6. Epress cos sin, where R > and 9, giving the eact value of R and the value of correct to decimal places. Let cos - sin Rcos( cos sin Rcos cos Rsin sin Equate the coefficien ts of cos and sin Rcos...( Rsin...( Replace R in (: cos cos 67. ( Answer 7. Epress sin cos 9 in the form R sin, where R > and 9, giving the eact value of R and the value of correct to decimal places.. Page of Let 9sin -cos Rsin( 9sin cos Rsin cos Rcos sin Equate the coefficien ts of cos and sin Rcos 9...( Rsin...( ( ( : R R cos R sin Replace R in (: cos cos 6 R 6 ( Answer 6. ( Answer 6 ( ( : R R cos R sin R ( Answer Replace R in (: 9 cos 9 cos. ( Answer Trigonometry Functions Page
. Prove the identity sec sec tan sin sec sec tan sin. cos cos cos sin cos cos sin cos sin - sin sin ( sin ( sin sin ( Shown 9. Prove the identity cot cot cose cot cot tan tan tan tan tan tan tan ( tan tan tan tan tan tan tan tan tan tan ( Shown tan. Show that tan tan sec sec sec sec cos cos cos cos cos cos sec cos ( Shown. Show that, sin is negative for all values of. cos sin cos ( sin ( sin ( cos ( cos Page of ( cos cos (cos ( cos ( cos cos ( ( cos ( cos cos cos ( cos cos sin cos tan Since tan for all, sin cos sin cos for all ( Shown cos cos Trigonometry Functions Page
. a Prove that 6. i Show that the equation sin osec sec c cos tan(6 tan(6 k can be b i Solve for the equation written in the form sin cosec sec ( tan k ( tan ii Hence solve the equation ii Find the eact value of cosec sec tan(6 tan(6 giving all solutions in the interval a sin cos ec sec i sin.sin tan(6 tan(6 k sin cos tan6 tan tan6 tan k cos sin tan6 tan tan6 tan (sin cos (sin cos sin cos tan tan k tan tan cos sin sin cos sin cos ( tan ( tan ( tan ( tan k ( tan ( tan cos sin cos ( Shown b i cos cos.7.,.6.7, 9. ii sin cos ec sec ( Answer cos cos cos ec sec sin cos cos ec sec sin ( Answer tan k tan ( tan k( tan ( Shown ii tan(6 tan(6 From part(i, k ( tan ( tan ( tan ( tan tan 9 tan tan tan tan Page of 6.,6. ( Answer 7. Given the following triangle. Prove that tan A tan B tan C tan Atan B tan C, A B C A ( B C tan tan( B C tan A tan tan( B C tan( B C tan A tan A tan( B C tan B tan C tan A tan B tan C tan A( tan B tan C tan B tan C tan A tan Atan B tan C tan B tan C tan A tan B tan C tan Atan B tan C ( Shown Trigonometry Functions Page