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1 . 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
2 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
3 . Solve each of the following equations giving all solutions in the given interval a cos 7.7, for 6. cos 7.7 cos , , ( ,6.9 ( Answer b 7sin( 6.9, for 6. 7sin( 6.9 sin( ,.. 6.9,, ,6. ( Answer c sin( 67., for d 6 cos(., for 6 sin( 67. sin( ,. 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
4 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
5 . 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
6 . 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
7 . 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
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