Trigonometry: Ratios, Identities & Trigonometric Table

Trigonometric ratios refer to the specific measurement methods of the right-angle triangle which has an interior angle as \(90^{\circ}\). Trigonometry helps to calculate and find the measurement of the sides of lengths or angles or both of a right-angle triangle. The various trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). In trigonometric ratios, the reference angle is denoted by \(\theta\).

Key Takeaways: Trigonometry, Trigonometric Ratios, Trigonometric Identities, Complementary angles, Right angle triangle

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Trigonometric Ratios

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Let us look at both cases:

Trigonometric Ratios Relation

The relation between different trigonometric relation is given below.

\(cosec \theta = 1/\sin \theta \)

\(\sec \theta = 1/\cos \theta\)

\(\tan \theta = \sin \theta/\cos \theta\)

\(\cot \theta = \cos \theta/ \sin \theta = 1/\tan \theta\)

Also Read:

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Trigonometric Ratios Range and Variation from 0 to \(90^{\circ}\) Angles

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Given below are the trigonometric ratios at specific and \(90^{\circ}\)angles

Range of Trigonometric Ratios between 0 to \(90^{\circ}\) Angles

Trigonometric Ratios range between 0 to 90 degrees:

For 0 degrees \(\leq\) \(\theta\) \(\leq\) 90 degrees

• \(0\leq\sin\theta\leq1\)
• \(0\leq\cos\theta\leq1\)
• \(0\leq\tan\theta<\infty\)
• \(1\leq\sec\theta<\infty\)
• \(0\leq\cot\theta<\infty\)
• \(1\leq cosec \theta<\infty\)

sec\(\theta\) and tan\(\theta\) are not defined at 90 degrees.

cosec\(\theta\) and cot\(\theta\) are not defined at 0 degrees.

Trigonometric Ratios Variation from \(0^{\circ}\) to \(90^{\circ}\)

Variation of the \(\theta\) increasing from 0 to 90 degrees:

• sin\(\theta\) increases from 0 to 1
• cos\(\theta\) decreases from 1 to 0
• tan\(\theta\) increases from 0 to \(\infty\)
• cosec\(\theta\) decreases from \(\theta \) to 1
• sec\(\theta\) increases from 1 to \(\infty\)
• cot\(\theta\) decreases from \(\infty\) to 0

Trigonometric Ratios at 90 degrees

Trigonometric ratios at \(90^{\circ}\) angle is \(90^{\circ} - \theta\) when \(\theta \) is an acute angle. 

• \(\sin (90^{\circ}-\theta) = cos \theta\)
• \(\cos (90^{\circ}-\theta) = sin \theta\)
• \(\tan (90^{\circ}-\theta) = \cot \theta\)
• \(\cot (90^{\circ}-\theta) = \tan \theta\)
• \(cosec (90^{\circ}-\theta) = \sec \theta\)
• \(\sec (90^{\circ}-\theta) = cosec \theta\)

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Trigonometric Identities

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The equation which consists of a trigonometric ratio of angle(s) is defined as a trigonometric identity. Depending, if it is real for all figures of the angles involved. The identities are mentioned below:

• tan θ = sinθ/cosθ
• cot θ = cosθ/sinθ
• sin² θ + cos² θ = 1 
• sin² θ = 1 – cos² θ
• cos² θ = 1 – sin² θ
• cosec² θ – cot² θ = 1 
• cosec² θ = 1 + cot² θ
• cot² θ = cosec² θ – 1
• sec² θ – tan² θ = 1 
• sec² θ = 1 + tan² θ 
• tan² θ = sec² θ – 1
• sin θ cosec θ = 1 
• cos θ sec θ = 1 
• tan θ cot θ = 1

Standard Value of Trigonometric Ratios

Values of standard angles of trigonometric ratios are given below.

Also Read:

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Points to Remember

1. A trigonometric ratio is purely based upon the angle ‘θ’ and remains the same for the same angle-sized right triangles which are different.
2. Besides this, the value of sin θ and cos θ will never exceed one as the opposite side is also 1.
3. The adjacent side is never greater than the hypotenuse since the hypotenuse is the longest side in a right-angled triangle.
4. In trigonometric ratios, sin\(\theta\) is equal to the opposite side by hypotenuse whereas cosec\(\theta\) is equal to hypotenuse by the opposite side.
5. cos\(\theta\) is equal to the adjacent side by the hypotenuse whereas sec\(\theta\) is equal to hypotenuse by the adjacent side.
6. tan\(\theta\) is the opposite side by the adjacent side whereas cot\(\theta\) is equal to the adjacent side by the opposite side.