ASTC Trigo Rule – How to Solve Trigo Equations

In this revision note, you will learn about the ASTC Trigo rule and how to use it to solve Trigonometry (Trigo) equations in O-Levels A-Math exams.

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Cartesian Plane

The cartesian plane can be divided into 4 equal parts. Each part is called a quadrant. We name each of the quadrant as shown in Figure 4 below.

trigo quadrants

 

Basic Angle or Reference Angle

Corresponding to each angle formed by the rotation of OP about the origin O, a unique acute angle (also commonly called the basic angle or reference angle or α) and a right-angled triangle OPQ can be observed, where Q is always on the x-axis such that ∡ is 90°. The figure below shows the relationship between the positive angle θ and basic angle α for each quadrant.

basic angle, trigo

                         θ in 2nd Quadrant                                                                θ in 1st Quadrant 

                         θ = 180° – α                                                                           θ = α

3rd and 4th quadrant

                       θ in 3rd Quadrant                                                                        θ in 4th Quadrant 

                       θ = –( 180° +  α)                                                                           θ = – (360° – α)

Negative Angle

The figure below shows the relationship between the negative angle θ and basic angle α for each quadrant

negative angle

                         θ in 2nd Quadrant                                                                     θ in 1st Quadrant 

                         θ = –( 180° + α)                                                                        θ = –( 360° – α)

                       θ in 3rd Quadrant                                                                    θ in 4th Quadrant 

                       θ = –( 180° – α)                                                                       θ = – α

ASTC Trigo Rule

To remember which trigonometric ratios are positive, your can follow the ASTC Trigo rule: ASTC (Add Sugar To Coffee).

astc trigo rule

The figure below explains the ASTC Trigo rule and why the trigonometric ratios are positive in each quadrant.

astc trigo rule, 1st and 2nd quadrant

                    θ in 2nd Quadrant

                    θ = 180° – α 

                  sin θ = sin α = PQ/OP    

                  cos θ = cos α = –OQ*/OP  

                  tan θ = tan α = –PQ/OQ*   

                 *OQ is negative in the value

                  since it is on the negative    

                  side of the x-axis. 

                  Only sine ratio in this         

                  quadrant is positive

 

θ in 1st Quadrant 

θ = α

sin θ = sin α = PQ/OP

cos θ = cos α = –OQ/OP

tan θ = tan α = –PQ/OQ

OQ and PQ are on the

positive sides of the axes

The hypotenuse is always

positive in each quadrant.

All trigonometric ratios in 

this quadrant are positive.

 

astc trigo rule

                    θ in 3rd Quadrant

                    θ = 180° +  α 

                  sin θ = sin α = – PQ* /OP    

                  cos θ = cos α = –OQ*/OP  

                  tan θ = tan α = PQ*/OQ*   

                 *OQ is negative in the value

                  since it is on the negative 

                  *PQ is negative in value

                   since it is on the negative

                   side of the y-axis.

                   side of the x-axis. 

                   Only tangent ratio in this         

                   quadrant is positive since

                   both PQ and OQ are 

                   negative.

θ in 4th Quadrant 

θ = 360° – α

sin θ = sin α = –PQ*/OP

cos θ = cos α = OQ/OP

tan θ = tan α = –PQ*/OQ

* PQ is negative in value since it is on 

the negative side of the y-axis.

Only cosine ratio in this quadrant

is positive.

 

 

 

 

 

 

 

Example of Solving Trigo Equations

Solve the following equations for 0° ≤ x ≤ 360°.

(a) cos x = 0.1256 (b) sin x = −1/2 (c) 2 tan x + 3 = 0

 

Solution:

(a)

cos x = 0.1256

Basic ∡ = 82.7846°

x = 82.7846°, 360° − 82.7846°

= 82.8°, 277.2° (1 dp) (ans)

astc trigo rule

Since cos x has a positive value, we infer x must be in the 1st or 4th Quadrant.

We will calculate the value of x for each of those quadrants.

In the first quadrant, x is simply the value of the basic angle α.

 

(b)

sin x = −1/2

Basic ∡ = 30°

x = 180° + 30°, 360° − 30°

= 210°, 330° (ans)

astc trigo rule

Since sin x has a negative value, we infer x must be in the 3rd or 4th Quadrant.

We will calculate the value of x for each of those quadrants.

Note that the calculation of α involves sin−1(1/2). Leave out the minus sign in the calculator since α is always acute.

 

(c)

2 tan x + 3 = 0

tan x = −3/2

Basic ∡ = 56.3100°

x = 180° − 56.3100°, 360 − 56.3100°

= 123.7°, 303.7° (1 dp) (ans)

astc trigo rule

Since tan x has a negative value, we infer x must be in the 2nd or 4th Quadrant.

We will calculate the value of x for each of those quadrants.

Note that the calculation of α involves tan−1(3/2). Leave out the minus sign in the calculator since α is always acute.

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