Fundamental Trigonometric Identities
Trigonometric identities are equations that relate various trigonometric functions and are true for any variable value in the domain. In this post, you can learn fundamental trigonometric identities.
Tutor-style math help
Fundamental Trigonometric Identities: what to notice and how to work it
Trigonometry skill
Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.
What to notice first
Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.
Common student mistake
Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.
Key formulas and cues
\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\sin^2\theta+\cos^2\theta=1\)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
Example: opposite = 5, hypotenuse = 13
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Answer: \(\sin\theta=\frac5{13}\)
Unit-circle cosine
Example: \(\cos(0)\)
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Answer: \(1\)
Try one before moving on
Try: In a right triangle, tangent equals which ratio?
Answer: Opposite over adjacent.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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Fundamental Trigonometric Identities: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Cosine in a right triangle is:
2. \(180^\circ\) equals:
3. The unit circle has radius:
A step-by-step guide to fundamental trigonometric identities
The basic trigonometric identities or fundamental trigonometric identities are those trigonometric functions that are true every time for the variables.
The following equations are eight of the most basic and important trigonometric identities. These equations are true for any angle. Countless additional identities can be formed from them. These eight things should be kept in mind.
- \(\color{blue}{cot\left(θ\right)=\frac{cos\:\left(\theta \right)}{sin\:\left(\theta \right)}}\)
- \(\color{blue}{tan\:\left(\theta \right)=\frac{sin\:\left(\theta \right)}{cos\:\left(\theta \right)}}\)
- \(\color{blue}{cot\left(θ\right)=\frac{1}{tan\:\left(\theta \right)}}\)
- \(\color{blue}{sec\left(θ\right)=\frac{1}{cos\:\left(\theta \right)}}\)
- \(\color{blue}{csc\left(θ\right)=\frac{1}{sin\:\left(\theta \right)}}\)
- \(\color{blue}{\left(sin\left(θ\right)\right)^2+\left(cos\left(θ\right)\right)^2=1}\)
- \(\color{blue}{1+\left(tan\left(θ\right)\right)^2=\left(sec\left(θ\right)\right)^2\:\:}\)
- \(\color{blue}{1+\left(cot\left(θ\right)\right)^2=\left(csc\left(θ\right)\right)^2}\)
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