Trigonometry: Sine, Cosine, Tangent
Trigonometry uses sine, cosine and tangent to find missing sides and angles in right-angled triangles. SOH CAH TOA picks the right ratio, such as sin θ = opposite ÷ hypotenuse. For a given angle the ratio never changes.
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What Are sin, cos, and tan?
- These are ratios used in right-angled triangles
- Once the angle is given, the ratios are fixed
Using sin, cos, and tan
- SOH:
- CAH:
- TOA:
Finding a Missing Side
- Choose sin, cos, or tan based on the sides you know and the side you want
- Substitute the values and rearrange to find the unknown side
Finding a Missing Angle
- Write the correct trig ratio first (sin, cos, or tan)
- Use the inverse function on the calculator (, , or )
Common Trigonometric Values
- You should know the exact values for , , and without a calculator
- Having these triangles in your head helps you see the ratios instantly
Practice Questions
Test your understanding
In a right-angled triangle, the opposite side is , and the hypotenuse is . What is the sine of the angle?
Correct! 🎉 +10 pointsNot quite right
The sine formula is . Here, the opposite side is and the hypotenuse is . So, the sine of the angle is .
In a right-angled triangle, the adjacent side is , and the hypotenuse is . What is the cosine of the angle?
Correct! 🎉 +10 pointsNot quite right
The cosine formula is . Here, the adjacent side is and the hypotenuse is . So, the cosine of the angle is .
What is the sine of the angle?
Correct! 🎉 +20 pointsNot quite right
The sine formula is . Here, the opposite side is and the hypotenuse is . So, the sine of the angle is .
What is the tangent of the angle?
Correct! 🎉 +20 pointsNot quite right
Using the tangent formula , here the opposite side is and the adjacent side is . So, the tangent of the angle is .
What is the cosine of the angle?
Correct! 🎉 +20 pointsNot quite right
Using the cosine formula , here the adjacent side is and the hypotenuse is . So, the cosine of the angle is .
Find the angle in the right-angled triangle.
Correct! 🎉 +30 pointsNot quite right
The sine formula is . Here, the opposite side is and the hypotenuse is , so . Using the inverse sine function, .
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Interactive Activity
Explore trigonometric ratios in a right-angled triangle
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Students Also Ask
The questions students bump into most on this topic
SOHCAHTOA is a memory aid for the three trigonometric ratios in a right-angled triangle. SOH means sine equals opposite over hypotenuse. CAH means cosine equals adjacent over hypotenuse. TOA means tangent equals opposite over adjacent, where the angle is labelled θ.
No. SOHCAHTOA only works in right-angled triangles, never in other triangles. Its three ratios, sine, cosine and tangent, are defined using the opposite, adjacent and hypotenuse sides. For triangles without a right angle, you would need different methods that are not covered here.
Choose the ratio that matches the two sides you know or want. Use sine for the opposite and hypotenuse. Use cosine for the adjacent and hypotenuse. Use tangent for the opposite and adjacent. First label the sides relative to the angle θ, then pick the matching ratio.
No. For a given angle θ, sine, cosine and tangent stay the same. This is true no matter how large or small the right-angled triangle is. Their values depend only on the angle, not on the side lengths. So any triangle with that angle gives the same ratios.
Remembering a few special triangles is often easier than memorising the values directly. These triangles store the exact sine, cosine and tangent values for common angles. You can then read off a value such as sin 30° = 1/2 instead of recalling it from memory.