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Trigonometry - calculating the value of a trig ratio given details of another.
Required strategies.


 

A very common Trigonometry question in Mathematics provides details of a trigonometric ratio and (either directly or indirectly) details about a second ratio. You have to calculate the value of a third ratio (which may be an otdirary ratio or a reciprocal relationship.

Remember the following steps. As an example we use the following typical question:

If sin θ = 1/2 and cos θ < 0 find the value of tan θ.

Step 1: Draw a right-angled triangle and mark it with the information given about the angle and the relevant ratio between the sides.

sin θ = 1/2.

So the opposite side is 1 and the hypotenuse is 2.
Step 2: Calculate the length of the third side using Pythagoras and add that to the diagram. The third side here is the adjacent side and its length is√3.
Step 3: Next to the first diagram, draw a pair of vertical and horizontal lines to show the 4 quadrants in a circle. Mark in ASTC.  
Step 4: Tick the quadrants showing the sign of this ratio for the given angle. The sine ratio for the given angle is positive and so ticks are made in the quadrants where sin is positive - so in the 1st and 2nd quadrants.
Step 5: Tick the relevant quadrants relating to the second piece of information about the given angle. The cos ratio is given as negative so add ticks to the two quadrants where cos is negative - so to the 2nd and 3rd quadrants.
Step 6: Select the quadrant having the two ticks.   Quadrant 2 has 2 ticks so that is where the required angle is located.
Step 7: Write down the ratio for the given angle using the trig ratio in the question and the diagram in Step 2.   tan θ = (√3)/2
Step 8: Write down the +ve or -ve sign of the ratio for this trig relationship using the quadrant marked with the two ticks.  

In quadrant 2, tan is negative and so we add a negative sign to the value for the previous step.

Hence the answer is tan θ = -(√3)/2