Table of Content

• What are Allied Angles?
• Formula of Trigonometric Ratios of Allied Angles
• Common T-Ratio of Allied Angles
• Conclusion
• Examples on Allied Angles
• Practice Problems on Allied Angles
• FAQs on T-Ratio of Allied Angles

Angle (θ)	-θ	(90° – θ)	(90° + θ)	(180° – θ)	(180° + θ)	(270° – θ)	(270° + θ)
sin θ	-sin θ	cos θ	cos θ	sin θ	– sin θ	– cos θ	– cos θ
cos θ	cos θ	sin θ	-sin θ	-cos θ	– cos θ	– sin θ	sin θ
tan θ	-tan θ	cot θ	-cot θ	– tan θ	tan θ	cot θ	– cot θ
cosec θ	-cosec θ	sec θ	cosec θ	cosec θ	– cosec θ	– sec θ	– sec θ
sec θ	sec θ	cosec θ	-cosec θ	-sec θ	– sec θ	– cosec θ	– cosec θ
cot θ	-cot θ	tan θ	-tan θ	– cot θ	cot θ	tan θ	– tan c

One trick that can be used to reduce any angle to a smaller values is that you should divide the angle by 90 and check for the quotient and remainder. Then you should divide the quotient by 4 and check for the quotient obtained.

Below is a list which describes meaning of each remainder:

• 0: Equivalent to 0 Degrees
• 1: Equivalent to 90 Degrees
• 2: Equivalent to 180 Degrees
• 3: Equivalent to 270 Degrees

Larger angle can be replaced as sum of remainder and above multiples of 90° as per the remainder obtained. Let us understand the above trick with help of an example,

Suppose, we need to find measure of sin (570°). Then, according to the trick, first we divide 570 by 90, we get quotient as 6 and remainder as 30. So, sin (570°) can be written as,

sin (570°) = sin (90°×6 + 30°)

Now, we divide 6 by 4, and get 2 as remainder, 2 corresponds to 180° as shown above. Thus,

sin (570°) = sin (180° + 30°)

Now, from the table, we see that sin (180° + θ) = – sin θ. Hence,

sin (570°) = -sin (30°) = -1/2

Thus, we can break the larger angles to smaller angles whose trigonometrical ratio value is known and get values for them.

Common T-Ratio of Allied Angles

Some common t-ratios formulas are listed below:

• sin (90 – θ) = cos θ
• cos (90 – θ) = sin θ
• tan (90 – θ) = cot θ
• sin (90 + θ) = cos θ
• cos (90 + θ) = -sin θ
• tan (90 + θ) = -cot θ

Conclusion

In conclusion, we have seen that the concept of allied angles is a crucial one in trigonometry that is predominantly used to determine the value of T-ratio for larger angle, if value for smaller angle is known by breaking the larger angle as a multiple of 90 degrees and the remainder being the smaller angle.

Allied angles find applications in various theoretical and practical fields such as determining the behaviour of T-ratios across the quadrants, wave motions, harmonic motions, signal processing, etc.

Examples on Allied Angles

Example 1: Find the value of sin(210°).

Solution:

We have,

⇒ sin (210°) = sin(180°+30°)

From table, we know that, sin(180°+θ) = -sin θ,

⇒ sin (210°)

= – sin (30°)

= -1/2

Thus, we get the value of sin (210°) as 1/2.

Example 2: Find the value of cot(330°).

Solution:

We have,

⇒ cot (330°)

= cot(270°+60°)

From table, cot(270° + θ) = -tan θ, we get,

⇒ cot (330°)

= -tan 30°

= -√3

Hence, we obtained the value of cot (330°) as -√3.

Practice Problems on Allied Angles

P1. What is the value of sin (240°)?

P2. Find the value of cos (120°).

P3. Find the value of tan (150°).

P4. Determine the value of cosec (570°).

P5. Determine the value of sec (720°).

FAQs on T-Ratio of Allied Angles

What is meant by allied angles?

Allied angles are a pair of angles whose sum or difference is a multiple of 90 degrees. For example, 150° and 30° form a pair of allied angles.

What are practical examples of allied angles?

Allied angles are very common in geometry. Their examples include complementary angles (angles that sum up to 90°), supplementary angles (angles that sum up to 180°), linear pair angles (angles formed on a line), co-interior angles (angles that lie on the same side of a transversal), etc.

How is the T-ratio of allied angles related?

Using the properties of allied angles, one can break a larger angle to a smaller angle, and determine value of the T-ratio for the larger angle. For example, sin (90°+θ) can be written as -cos θ and in this way we were able to break the (90°+θ) to θ, for which the value of T-ratio is known.

What are practical applications of allied angles?

Allied angles have applications in various fields such as wave motion, harmonic motion (as they involve periodicity), signal processing, structural analysis, and navigation, etc.

Can allied angles be negative?

Yes, allied angles can be negative. Only condition that is required to be met is that their sum or difference must be a multiple of 90 degrees.