22. Trigonometry and standard values
Free study material · concepts, shortcuts & solved questions
In a right triangle, sinθ=P/H, cosθ=B/H and tanθ=P/B. Remember reciprocal ratios: cosec=H/P, sec=H/B, cot=B/P. Identities include sin²θ+cos²θ=1 and 1+tan²θ=sec²θ. Standard values at 0°,30°,45°,60°,90° should be learned as a table.
Method before formula Draw the triangle and label opposite, adjacent and hypotenuse relative to the given angle. Heights and distances questions are usually simple right-triangle applications; check whether the angle is elevation or depression. |
Formula and decision table
θ | sin, cos, tan |
0° | 0, 1, 0 |
30° | 1/2, √3/2, 1/√3 |
45° | 1/√2, 1/√2, 1 |
60° | √3/2, 1/2, √3 |
90° | 1, 0, undefined |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
If P:B:H=3:4:5, sinθ: | sin=P/H=3/5. | B. 3/5 |
tan45° equals: | tan45°=1. | C. 1 |
sin²θ+cos²θ equals: | Fundamental identity. | B. 1 |
What an examiner is testing
The options around Trigonometry and standard values usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: If P:B:H=3:4:5, sinθ: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 3/5, because sin=P/H=3/5.
Example 2: tan45° equals: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 1, because tan45°=1.
Example 3: sin²θ+cos²θ equals: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 1, because Fundamental identity.
Chapter practice
1. If P:B:H=3:4:5, sinθ:
(A) 3/4 (B) 3/5 (C) 4/5 (D) 5/3
Answer: B. 3/5 | Explanation: sin=P/H=3/5.
2. tan45° equals:
(A) 0 (B) 1/2 (C) 1 (D) √3
Answer: C. 1 | Explanation: tan45°=1.
3. sin²θ+cos²θ equals:
(A) 0 (B) 1 (C) 2 (D) tanθ
Answer: B. 1 | Explanation: Fundamental identity.
SSC CGL speed and trap clinic
For Trigonometry and standard values, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Trigonometry and standard values? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Trigonometry and standard values question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |