higher mathematics – vectors€¦ · web view(c)(d) (e)(f) (g)(h) (i)(j) 2.sketch each graph...

Higher Mathematics – Vectors

Education Resources

Trigonometry

Higher Mathematics Supplementary Resources

Section A

This section is designed to provide examples which develop routine skills necessary for completion of this section.

R1 I can convert radians to degrees and vice versa.

1.Convert the following angles from degrees to radians, giving you answer as an exact value.

(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)

(n)(o)(p)

2.Convert the following angles from degrees to radians, giving you answer to 3 significant figures.

(a)(b)(c)

(d)(e)(f)

3.Convert the following angles from radians to degrees.

(a) radians(b) radians(c) radians

(d) radians(e) radians(f) radians

(g) radians(h) radians(i) radians

(j) radians(k) radians(l) radians

(m) radians(n) radians(o) radians

(p) radians(q) radians(r) radians

(s) radians(t) radians(u) radians

4.Convert the following angles from radians to degrees, giving you answer to 3 significant figures.

(a) radian(b) radians(c) radians

(d) radians(e) radians(f) radians

R2 I can use and apply exact values.

1.Write down the exact value of

(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)


(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)


(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)


(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)


(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)


(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(n)(o)

R3I can sketch or identify a basic trig graph under a single transformation.

1.Write down the equation of each of the graphs

(a)(b)

(c)(d)


(a)(b)

(c)(d)


(a)(b)

(c)(d)


(a)(b)

5.Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes.

(a) for (b) for

(c) for (d) for

(e) for

(f) for

(g) for (h) for

R4I can sketch or identify a basic trig graph under combined transformations.


(a)(b)

(c)(d)

(e)(f)

(g)(h)

(i)(j)

2.Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes.

(a) for (b) for

(c) for

(d) for

(e) for

(f) for

(g) for (h) for

R5I can use the addition and double angle formulae.

1.Expand and use exact values to simplify

(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

2.Use an appropriate substitution (such as 45 – 30 = 15) then expand to find the exact values of

(a)(b)(c)

3.Given that and , find the exact values of:

(a)

(b)

(c) (Hint )


(a)

(b)

(c) (Hint )


(a)

(b)

(c)


(a)

(b)

(c)

R6 I can convert to or where α is in any quadrant .

1.A function is defined as .

Express in the form where and .

2.Express in the form where and .












R7I have revised solving basic trigonometric equations in degrees and radians.

1.Solve the equations:

(a)

(b)

(c)

(d)

(e)

(f)


(a)

(b)

(c)


(a)

(b)

(c)

(d)

(e)

(f)

Section B

This section is designed to provide examples which develop Course Assessment level skills

( )NR1 I can apply Trig Formulae to Mathematical Problems (excluding where trig equations have to be solved but including exact values).

1.On the coordinate diagram shown, P is the point and Q is the point .

Angle POR = and angle ROQ = .

(a)Find the exact value of .

(b)Find the exact value of .

(ACB)

2.In triangle ABC, show that:

(a)The exact value of

(b)The exact value of (A)

3.For the shape shown, find the exact value of

(ABC25)

(PQR)

4.The diagram shows the right angled triangle PQR, with dimensions given.

(a)Find the exact value of .

(b)By expressing as , find the exact value of .

5.If and , find the exact values of and .

6.Using the fact that , find the exact value of .

7.It is given that and .

(a)Find the exact value of and .

(b) (A)Hence find the exact value of .

(PQRS)

8.Triangles PSQ and RSQ are right angled with dimensions as shown in the diagram.

(a)Show that is .

(b) (A)Calculate the value of .

(c)Hence calculate the value of .

NR2 I have experience of using wave functions to find the maximum and minimum values.


(a)Express in the form where and .

(b)Part of the graph of is shown in the diagram.

Find the coordinates of the minimum turning point A.

2.(a)Express in the form where and .

(b)Hence state the maximum and minimum values of and determine the values of , in the interval , at which these maximum and minimum values occur.


(b)Hence state the maximum value of and determine the value of , in the interval , at which the maximum occurs.


Find the maximum and minimum values of and the values of, in the range , at which the maximum and minimum values occur.


Find the maximum and minimum values of and the values of, in the range , at which the maximum and minimum values occur.

NR3 I can solve trigonometric equations in the context of a problem.

1.Solve the equation , in the interval .








(b)Hence solve the equation in the interval .


(b)Hence solve the equation in the interval .

10.Two curves have equations and .

Find the coordinates of the points of intersection in the range .

11.Two curves have equations and .

Find the coordinates of the points of intersection in the range .

12.A curves has the equation .

Find the coordinates of the points where the curve cuts the -axis in the range .

13.A curves has the equation .

Find the coordinates of the points where the curve cuts the -axis in the range .

14.The graph shows two curves which have equations and in the range .

(AB)

Find the coordinates of A and B, the points of intersection between the two curves.

NR4 I have experience of cross topic exam standard questions.

Trigonometry and integration

1.(a)The expression can be written in the form where and .

Calculate the values of and .

(b)Hence find the value of for which

.

2.A curve for which passes through the point .

Find in terms of .

3.Find the area enclosed by and .

Trigonometry and differentiation

1.Two functions are defined as and .

(a)Write in the form where and .

(b)Hence find as a single trigonometric expression.

2.A curve has equation .

(a)Write in the form where and .

(b)Hence find, in the interval , the -coordinate of the point on the curve where the gradient is 1.

3.Find the equation of the tangent to the curve at the point where

Trigonometry, functions and graphs


(a)Express in the form where and .

(b)Sketch the graph of between , showing clearly the coordinates of the maximum and minimum turning points.


(b)Sketch the graph of between , showing clearly the coordinates of the maximum and minimum turning points and where the curve cuts the axes.

3.Functions , and are defined on a suitable set of real numbers.

(a)Find expressions for;

(i);

(ii).

(b)(i)Show that .

(ii)Find a similar expression for and hence solve the equation for .

4.Functions and are defined on suitable domains by and .

(a)Find expressions for;

(i);

(ii).

(b)Solve for .

Trigonometry and straight line

((6, 5))1.P is the point . The line is inclined at an angle of to the -axis.

(a)Find the exact values of and .

(b)The line is inclined at an angle of to the -axis. Write down the exact value of the gradient of .

Answers

R1

1.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)(p)(q)(r)

2.(a)(b)(c)(d)(e)(f)

3.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)(p)(q)(r)

(s)(t)(u)

4.(a)(b)(c)(d)(e)(f)

R2

1.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)

2.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)

3.(a)(b)(c)(d)(e)(f)

(g) Undefined(h)(i) Undefined(j)(k)(l)

(m)(n)(o)

4.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)

5.(a)(b)(c)(d)(e)(f)

(g)(h)(i)(j)(k)(l)

(m)(n)(o)

6.(a)(b)(c)(d)(e)(f)Undefined

(g) (h)(i)(j)(k)(l)

(m)(n)(o)

R3

1.(a)(b)(c)

(c)

2.(a)(b)(c)

(c)

3.(a)(b)(c)

(c)

4.(b)(c)

5.Sketches can be check by using a graph drawing package

R4

1.(a)(b)(c)

(d) or (e)

(f)(g)(h)

(i)(j)

2.Sketches can be check by using a graph drawing package

R5

1.(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

2.(a)(b)(c)

3.(a)(b)(c)

4.(a)(b)(c)

5.(a)(b)(c)

6.(a)(b)(c)

R6

1.2.3.

4.5.6.

7.8.9.

R7

1.(a)(b)(c)

(d)(e)(f)

2.(a)(b)

(c)

3.(a)(b)(c)

(d)(e)(f)

NR1

1.(a)(b)

2.(a)Proof(b)Proof3.

4.(a)(b)5.,

6.(a)7.(a),

7.(b) 8.(a)Proof(b)(c)

NR2

1.(a)(b)

2.(a)(b)min at , max at

3.(a)(b) max at

4.min at , max at

5.min at , max at

NR3

1.2.3.

4.5.6.

7.8(a)(b)

9(a)(b)10.

11.12.13.

14. and

NR4

Trigonometry and integration

1(a)(b)2.

3.Area = square units

Trigonometry and differentiation

1(a)(b)

2(a).(b)3.

Trigonometry and functions and graphs

1(a)

(b)Sketches can be check by using a graph drawing package

2(a).

(b)Sketches can be check by using a graph drawing package

3(a).(i)(ii)

(b)(i)Proof(ii)

4(a).(i)(ii)

(b)

Trigonometry and the straight line

1(a),

(b)

Higher Mathematics – TrigonometryPage 1

SLC Education Resources – Calderglen High SchoolPage 16

Higher Mathematics – TrigonometryPage 15

higher mathematics – vectors€¦ · web view(c)(d) (e)(f) (g)(h) (i)(j) 2.sketch each graph...

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