TRIGONOMETRY

RADIANS.

e.g. if A is a full turn

Note: The area of the sector in the diagram is given by:

Questions:2003 II 4a, 2001 II 4a, 2000 II 4a, 1994 II 5a.

Harder Questions: 2005 II 4c, 2004 II 4c, 2003 II 4c, 2002 II 4c(iii),

RIGHT-ANGLED TRIANGLES

SOHCAHTOA and Pythagoras’ Theorem.

Questions: 2002 II 4c(i), (ii), 1994 II 5c.

ANGLES GREATER THAN 90º

The unit circle, x = (cosA, sinA) and ASTC

AREA FORMULA

Questions: 2005 II 5a, 2002 II 5a, 2001 II 5b (i), 1999 II 5a,

Harder Questions: 1998 II 4c(i),

THE GOLDEN RULE

This equation is used constantly to change to and vice versa.

It can be proved using a right-angled triangle with a hypotenuse of length one and Pythagoras’ theorem.

COMPOUND ANGLE FORMULAE (NUMERICAL QUESTIONS)

Questions: 1998 II 5b, 1996 II 5a

GIVEN SINA FIND COSA ETC.

EXAMPLE given find . (NB check if A is acute or obtuse!)
Draw a right-angled triangle with opposite = 2 and hypotenuse = 7. Use Pythagoras’ theorem to find the adjacent. Use SOHCAHTOA to write down the ratio for .

Questions 2004 II 4a(i), 1998 II 5a

SIN COS AND TAN OF SPECIAL ANGLES.

(a)

Use the triangle and SOHCAHTOA.

(b)

Use the triangle and SOHCAHTOA.

(c)

e.g. to find in surd form.

Note: page 9 of the tables is invaluable here.

(d) etc.
Use the double angle formula, e.g. (page 9, Tables).

Questions: 2003 II 5a, 2002 II 5b (ii), 1999 II 4a, 1999 II 4c (i) (ii),

CHANGING SUMS TO PRODUCTS AND VICE VERSA.

Questions: 2001 II 5c, 1999 II 4a, 1999 II 5c,

PROOFS

Prove that    2005 II 4b(i)
Prove    2003 II 5c(i)
                 Extension: Prove
Prove that    2002 II 5b(i), 1999 II 4c
                  Extension: Hence how that
                  Extension: Find, in the form
                                                (i) tan 75^o     (ii) tan 15^o.
Prove that    2004 II 5a
                  Extension: Show that simplifies to a constant.

Prove that    2001 II 4c
                  Extension : Show that
Derive the formula     2000 II 4c
                  Extension : Show that
Show that    2000 II 5b
                  Extension : Find values of the integers l and k so that


Cosine Rule:, 1997 II 5b,

ONE TERM TRIGONOMETRIC EQUATIONS

e.g where

Find an acute angle A that solves the equation
Then check and Only give those answers that satisfy the original equation.

Questions: 2002 II 4a, 1998 II 4a, 1997 II 4a

TRIGONOMETRIC EQUATIONS.

The basic idea of equations, finding a product and solving FIRST FACTOR = 0 and SECOND FACTOR = 0 will work here.

Look out for chances to use the double angle formulae, e.g. if one angle is x and the other is 2x or . Quite often these formulae will change a trigonometric equation into a quadratic.

If all the angles are the same, e.g. x, it may be possible to use the golden rule to express in terms of .

Let and factorise the resulting quadratic. Solve for y and hence x.

Questions:
   2005 II 4b(ii)
   2004 II 4b(ii)
    2003 II 4b
   2002 II 4b
   2001 II 4b
   (Change sum to product.) 2001 II 5c and
   2000 II 4b
   1999 II 4b
   1999 II 5c
    1998 II 4b
    1997 II 5b
1996 II 5c
   1995 II 5b
   1994 II 5b.