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Calculating All Trigonometric Ratios

Learn to calculate all six trigonometric ratios if one ratio is given by using the Pythagorean theorem.

7 micro-lessons

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Finding All Ratios from One

Explain that knowing one ratio means knowing two sides (as a ratio), allowing calculation of the third side.

If you know just one trigonometric ratio, you can unlock all the other five!

For example, if you are given $\sin A = \frac{1}{3}$ , this tells you the ratio of the side opposite to angle A to the hypotenuse is $1:3$ .

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The Step-by-Step Method

A clear, numbered algorithm for finding missing trigonometric ratios.

Key Takeaway: The 4-Step Method

To find all trigonometric ratios from a single given ratio, follow these steps:

Draw a right triangle and identify your reference angle.
Assign lengths to two sides based on the given ratio, multiplying each by a constant $k$ (e.g., $x k$ and $y k$ ).
Calculate the length of the missing third side using the Pythagorean Theorem.
Write down the requested trigonometric ratios using your three known sides, letting the $k$ values cancel out.

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Reference Triangle for Example 1

Visualizing the reference triangle constructed from a given ratio.

Clean scientific diagram of a right-angled triangle ABC. The right angle is at vertex B. The vertical side opposite to a…

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Example 1: Given one ratio, find others

Find other trigonometric ratios of angle A given tan A = 4/3.

Problem. Given $\tan A = \frac{4}{3}$ , find the other trigonometric ratios of the angle A.

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Example 3: Evaluating an expression

Determine values of cos^2(theta) + sin^2(theta) given side lengths.

Problem. Consider $\triangle ACB$ , right-angled at C, in which $AB = 29$ units, $BC = 21$ units and $\angle ABC = \theta$ .

Determine the value of (i) $\cos^2 \theta + \sin^2 \theta$ .

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Your Turn: Faded Example

Find cos A and tan A given sin A = 3/4.

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If $\sin A = \frac{3}{4}$ , we can set the opposite side to $3k$ and the hypotenuse to $4k$ . Using the Pythagoras Theorem, the adjacent side squared is $(4k)^2 - (3k)^2 = 16k^2 - 9k^2 = 7k^2$ . Taking the square root, the adjacent side is exactly . Now we can find the other ratios. Since cosine is adjacent over hypotenuse, $\cos A =$ . Since tangent is opposite over adjacent, $\tan A =$ .

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Check Your Calculation

Determine sin A given 15 cot A = 8.

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Given $15 \cot A = 8$ , find the value of $\sin A$ .

← previousDefining Trigonometric Ratios next →Trigonometric Ratios Of 30°, 45°, And 60...

content

Finding All Ratios from One

Explain that knowing one ratio means knowing two sides (as a ratio), allowing calculation of the third side.

If you know just one trigonometric ratio, you can unlock all the other five!

For example, if you are given $\sin A = \frac{1}{3}$ , this tells you the ratio of the side opposite to angle A to the hypotenuse is $1:3$ .

1 / 3

content

Example 1: Given one ratio, find others

Find other trigonometric ratios of angle A given tan A = 4/3.

Problem. Given $\tan A = \frac{4}{3}$ , find the other trigonometric ratios of the angle A.

1 / 5

content

Example 3: Evaluating an expression

Determine values of cos^2(theta) + sin^2(theta) given side lengths.

Problem. Consider $\triangle ACB$ , right-angled at C, in which $AB = 29$ units, $BC = 21$ units and $\angle ABC = \theta$ .

Determine the value of (i) $\cos^2 \theta + \sin^2 \theta$ .

1 / 6

reading

Your Turn: Faded Example

Find cos A and tan A given sin A = 3/4.

0 of 3 blanks filled

quiz

Check Your Calculation

Determine sin A given 15 cot A = 8.

1 / 5

Given $15 \cot A = 8$ , find the value of $\sin A$ .