Chapter 10: Trigonometric Identities And Equations

Question 1

What is a unit circle?

Answer 1

. A circle on a graph with a radius of 1 unit.
. Each quadrant on the graph has a quarter of the unit circle.
. The circle’s centre is at the origin.

Question 2

How can you use a unit circle to find the values of sine, cosine or tangent for any angle of ‘theta’?

Answer 2

. Put point P(x,y) on circumference of unit circle in first quadrant.
. Make right-angle triangle with vertices of point P, the origin (O) and point on x-axis that is directly below point P.
. Label angle at origin, between hypotenuse (radius) and x-axis line, ‘theta’.
. Label x-axis line ‘x’, label line between P and x-axis ‘y’ and label radius ‘1’ (for 1 unit).
. Cos (‘theta’)= ‘x’= x-coordinate of P (adjacent/hypotenuse).
. Sin (‘theta’)= ‘y’= y-coordinate of P (opposite/hypotenuse).
. Tan (‘theta’)= ‘y/x’= gradient of line OP (opposite/adjacent).
- Use 3 definitions above to find any value of sin/cos/tan ‘theta’.
- Always measure positive angles of ‘theta’ anti-clockwise from the positive x-axis and always measure negative angles of ‘theta’ clockwise from the positive x-axis.
- If ‘theta’ is more than 360°, then think of ‘theta’ as ‘theta - 360°’.

Question 3

Why is tan(90°) undefined (answer in terms of the unit circle)?

Answer 3

. Because the radius of the unit circle is in line is vertical therefore y=1 and x=0.
. tan (90°) = y/x = 1/0 = math error

Question 4

How do you determine which different trigonometric ratios are positive or negative in different quadrants?

Answer 4

. CAST (Cos, All, Sin, Tan) diagram.
. This diagram consists of a graph with 4 quadrants.
. For angle ‘theta’ in the 4th quadrant, only cos(‘theta’) is positive.
. For angle ‘theta’ in the 1st quadrant, sin/cos/tan (‘theta’) are positive.
. For angle ‘theta’ in the 2nd quadrant, only sin(‘theta’) is positive.
. For angle ‘theta’ in the 3rd quadrant, only tan(‘theta’) is positive.

Question 5

What is sin(180°-‘theta’) equivalent to?

Answer 5

Sin (‘theta’)

Question 6

What is cos(180°-‘theta’) equivalent to?

Answer 6

-cos(‘theta’)

Question 7

What is tan(180°-‘theta’) equivalent to?

Answer 7

-tan(‘theta’)

Question 8

What is sin(180°+’theta’) equivalent to?

Answer 8

-sin(‘theta’)

Question 9

What is cos(180°+’theta’) equivalent to?

Answer 9

-cos(‘theta’)

Question 10

What is tan(180°+’theta’) equivalent to?

Answer 10

tan (‘theta’)

Question 11

What is sin(360°-theta’) equivalent to?

Answer 11

-sin (‘theta’)

Question 12

What is cos(360°-theta’) equivalent to?

Answer 12

Cos (‘theta’)

Question 13

What is tan(360°-theta’) equivalent to?

Answer 13

-tan (‘theta’)

Question 14

How do you use triangles to find exact values of sin/cos/tan (30° or 60°)?

Answer 14

. Consider an equilateral triangle (ABC) of side lengths of 2 with side BC flat and sides AB and AC, meeting diagonally at vertex ‘A’ (angles inside triangle are 60°).
. Draw a vertical line from vertex ‘A’ and label other end of vertical line ‘D’ (midpoint of BC).
. This gives a right angled triangle of ABD (AB is hypotenuse).
. Angle ABD is 60°, angle BAD is 60°/2 = 30°, angle BDA is 90°.
. Side length AB is 2, side length BD is 2/2 = 1, side length AD is root(2²-1²) = root (3).
. Since you have all side lengths and the angles of 30° and 60° within a right-angled triangle, you can now find values of sin/cos/tan (30° or 60°) by using SOHCAHTOA.

Question 15

How do you use triangles to find exact values of sin/cos/tan (45°)?

Answer 15

. Consider an isosceles right-angled triangle (PQR) with PR being the hypotenuse.
. Side length PQ is 1, side length QR is 1, side length PR is root(1²+1²)=root(2).
. Angle PQR is 90°, angle PRQ is 45°, angle RPQ is 45°.
. Since you have all side lengths and the angle of 45° within an isosceles triangle, you can now find values of sin/cos/tan (45°) by using SOHCAHTOA.

Question 16

Why does sin²(theta)+cos²(theta)=1 (using unit circle)?

Answer 16

. Consider right-angled triangle in unit circle.
. Sin(theta)= length of y (the y-coordinate) ➗ 1 (length of hypotenuse) = length of y.
. Cos(theta)= length of x (the x-coordinate) ➗ 1 (length of hypotenuse)= length of x.
. Pythagoras theorem: y²+x²= 1 (because 1 is the length of hypotenuse/length of radius).
- Therefore sin²(theta)+cos²(theta)=1

Question 17

Why does sin(theta)/cos(theta) = tan(theta) (using unit circle)?

Answer 17

. Consider right-angled triangle in unit circle.
. Sin(theta)= length of y (the y-coordinate) ➗ 1 (length of hypotenuse) = length of y.
. Cos(theta)= length of x (the x-coordinate) ➗ 1 (length of hypotenuse)= length of x.
. tan(theta)= length of y (the y-coordinate) ➗ length of x (the x-coordinate)= y/x
. Therefore tan(theta)= sin(theta)/cos(theta)

Question 18

Give a formula similar to sin(theta)/cos(theta) = tan(theta):

Answer 18

sin²(theta)/cos²(theta) = tan²(theta)

Question 19

What important thing should you remember when square rooting a trig function (where you know roughly what value ‘theta’ is)?

Answer 19

Determine whether answer is positive or negative by considering whether the trig function is sin/cos/tan and whether ‘theta’ is acute/obtuse/reflex (think CAST diagram).

Question 20

What is the principal value (give the ranges for principal values)?

Answer 20

. The angle you get when you use inverse trigonometric functions on your calculator (for example sin^-1(theta) ).
. Principal values for ‘cos^-1(theta)’: 0

Question 21

Give another name for each inverse trigonometric function:

Answer 21

Arcsin, arccos, arctan

Question 22

What is the difference between the 2 meanings of ‘sin/cos/tan ^-1 (theta)’?

Answer 22

. 1st meaning: arcsin/arccos/arctan - these are inverse trigonometric functions.
. 2nd meaning: cosec/sec/cot- these are trigonometric functions that are to the power of -1 (1➗trigonometric function).
- Cosec represents sin, sec represents cos, cost represents tan (think of 3rd letter).