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How to find unknown angles in right triangles using the tangent ratio.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find unknown angles in right triangles using tangent ratios. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

For any ratio of two numbers, if both numbers in the ratio are multiplied by the same scale factor, then the ratio is equivalent. This is also true of the tangent ratio.Â Consider this diagram...

If the opposite and adjacent sides are multiplied by the same scale factor, then the angles in the triangle don't change even though the triangle is getting bigger, and tangent ratios do not change either.

Even though the opposite and adjacent sides get bigger,Â

 the tangent ratio: does not change.

Â So the value of tan 40º is always the same in triangles of any size!

This table shows several values of opposite and adjacent sides that all give the same tangent ratio:

 Opposite leg Adjacent leg tangent ratio 3 5 0.6 6 10 0.6 9 15 0.6 12 20 0.6 15 25 0.6

Try dividing the opposite side length by the adjacent side length and in every case the answer is 0.6

By constructing these triangles, we find that they all contain an angle of 30.96375653 degrees.

So:Â Â  tan 30.96375653° = 0.6

Note: The tangent ratio does have meaning for angles outside the range ofÂ  0º to 90º, but these angles do not apply to right angled triangles, and so will not be discussed here.

Tangent ratios used to be printed in books of tables. These days, most scientific calculators have a built in 'tan' function that calculates tangent ratios.

WARNING: When using the 'tan' function on a pocket calculator, make sure the angle mode is set to degrees unless you intend otherwise. This is often indicated by 'deg' on the display area.

If you know the tangent ratio of an angle, finding the unknown angle is called the 'inverse tan' operation, and is often written as: tan-1 or atan.

So:Â Â  tan-1(0.6) = 30.96375653°

Maths Helper Plus also has a 'tan' function built in. It is also able to display a helpful diagram.

### Â Method:

Maths Helper Plus can calculate the tangent or inverse tangents, showing working steps and a helpful diagram.

#### Step 2Â  Display the parameters box

Press the F5 key to display the parameters box:

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You enter the given information into these edit boxes as follows:

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edit box 'A' = an angle of the triangle

NOTE: Just type a number for the angle. Do NOT use the degree operator: ° after the angle.

edit box 'B' = the horizontal leg of the triangle. ( Leave set to 5 )

edit box 'D' = a tangent ratio value for which you need to know the angle.

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To find the tan of a given angle, and display a diagram:

Click on the 'A' edit box, type an angle in degrees, then click the 'Update' button.

The calculations for finding the tangent ratio are displayed in black text, and the diagram shows one possible triangle containing angle 'A'.

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To find an angle, given the tan of the angle:

Click on the 'D' edit box. Type the tangent ratio as a decimal fraction: eg 0.6, OR as a ratio of two sides, eg: 3/5. Click update to calculate the required angle. The calculations are shown in blue text. This feature does not change the diagram.

Example 1: Sketch a right triangle containing an angle of 30°, and calculate the tan of 30° by dividing the lengths of the opposite and adjacent sides.

Solution: Click on the 'A' edit box in Maths Helper Plus (As described above), type 30 and click the 'Update' button. The calculations will appear in black text, and a diagram will be displayed.

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Example 2: The tan of angle 'A' is 0.2679. What is angle 'A' ?

Solution: Click on the 'D' edit box in Maths Helper Plus (As described above), type 0.2679 and click the 'Update' button. The working and answer is displayed in blue text, with the angle represented by 'Q', as follows:

If tanQ° = 0.2679,

then Q° = inverse tan of 0.2679

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = 14.9974°

Rounding off, we have: A = 15°.

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Example 3: Find the unknown angle 'A' in the following triangle...

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Solution: The tangent ratio for angle 'A' is given by 'opposite' divided by 'adjacent' = 2.8433/4.9248. Click on the 'D' edit box in Maths Helper Plus (As described above), type 2.8433/4.9248 and click the 'Update' button. The working and answer is displayed in blue text, with the angle represented by 'Q', as follows:

If tanQ° = 0.577343,

then Q° = inverse tan of 0.577343

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = 29.9997°

Rounding off, we have: A = 30°.

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